<PAGE>   1
             =======================================================

                       SECURITIES AND EXCHANGE COMMISSION

                              Washington, DC 20549

                        ---------------------------------

                                    FORM 10-Q

 X   QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES EXCHANGE
---
     ACT OF 1934

For the quarterly period ended   January 31, 2001
                               -------------------------------------------------

     TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES
---
     EXCHANGE ACT OF 1934

For the transition period from _______________ to ______________________________

     Commission File Number    001-15167
                            ----------------------------------------------------


                               BIOPURE CORPORATION
             (Exact name of registrant as specified in its charter)


         Delaware                                       04-2836871
 (State of Incorporation)                   (IRS Employer Identification Number)

11 Hurley Street, Cambridge, Massachusetts                              02141
 (Address of  principal executive offices)                            (Zip Code)

                                 (617) 234-6500
                         (Registrant's telephone number)

                        ---------------------------------


Indicate by check mark whether the registrant (1) has filed all reports required
to be filed by Section 13 or 15(d) of the Securities Exchange Act of 1934 during
the preceding 12 months (or for such shorter period that the registrant was
required to file such reports), and (2) has been subject to such filing
requirements for the past 90 days. Yes X No _

The number of shares outstanding of each of the issuer's classes of common stock
as of February 28, 2001 was:

     Class A Common Stock, $.01 par value ..................24,956,197
     Class B Common Stock, $1.00 par value ......................117.7


             =======================================================

<PAGE>   2

                               BIOPURE CORPORATION

                               INDEX TO FORM 10-Q


                                                                            PAGE

Part I - Financial Information:

    Item 1 - Financial Statements (Unaudited)

           Condensed Consolidated Balance Sheets at
           January 31, 2001 and October 31, 2000                               1

           Condensed Consolidated Statements of Operations
           for the quarters ended January 31, 2001 and January 29, 2000        2

           Condensed Consolidated Statements of Cash Flows
           for the three months ended January 31, 2001 and January 29, 2000    3

           Notes to Condensed Consolidated Financial
           Statements                                                        4-6


    Item 2 - Management's Discussion and Analysis of
             Financial Condition and Results of Operations                   7-8

    Item 3 - Quantitative and Qualitative Disclosure of Market Risk            9


Part II - Other Information:

    Item 1 -   Legal Proceedings                                              10

    Item 2 -   Changes in Securities and Use of Proceeds                      10

    Item 6 -   Exhibits and Reports on Form 8-K                               10

Signatures                                                                    11

<PAGE>   3

                                                                       FORM 10-Q
                                                                       PART 1
                                                                       ITEM 1
                                                                       PAGE 1
                               BIOPURE CORPORATION

                      Condensed Consolidated Balance Sheets
                 (In thousands, except share and per share data)
                                   (Unaudited)

<TABLE>
<CAPTION>
Assets:                                                 January 31, 2001                    October 31, 2000
                                                        ----------------                    ----------------
<S>                                                       <C>                                 <C>
Current assets:
    Cash and cash equivalents                             $      77,929                       $     88,828
    Accounts receivable, net                                        421                                478
    Inventories, net                                              3,158                              2,726
    Current portion of restricted cash                            3,508                              3,508
    Other current assets                                            408                                380
                                                          -------------                       ------------
         Total current assets                                    85,424                             95,920

Property, plant and equipment, net                               24,997                             25,061
Investment in affiliate                                              66                                 66
Other assets                                                        219                                240
                                                          -------------                      -------------
         Total assets                                     $     110,706                       $    121,287
                                                          =============                       ============

Liabilities and stockholders' equity:
Current liabilities:
    Accounts payable                                      $         341                       $      1,764
    Accrued expenses                                              8,924                              9,228
                                                          -------------                       ------------
         Total current liabilities                                9,265                             10,992

Deferred compensation                                             1,803                              1,785

Stockholders' equity:
    Convertible preferred stock, $0.01 par value,
      30,000,000 shares authorized, no shares
      outstanding                                                     -                                  -
    Common stock:
      Class A, $0.01 par value, 100,000,000 shares
        authorized, 24,953,061 shares outstanding at
        January 31, 2001 and 24,937,995 at
        October 31, 2000                                            250                                249
      Class B, $1.00 par value, 179 shares
        authorized, 117.7 shares outstanding                          -                                  -
    Capital in excess of par value                              373,568                            372,149
    Contributed capital                                          24,574                             24,574
    Notes receivable                                             (1,724)                            (2,033)
    Accumulated deficit                                        (297,030)                          (286,429)
                                                          -------------                       ------------
         Total stockholders' equity                              99,638                            108,510
                                                          -------------                      -------------
             Total liabilities and stockholders'
          equity                                          $     110,706                       $    121,287
                                                          =============                       ============
</TABLE>


Note: The balance sheet at October 31, 2000 has been derived from the audited
financial statements at that date but does not include all of the information
and footnotes required by accounting principles generally accepted in the United
States for complete financial statements.
                             See accompanying notes.

<PAGE>   4

                                                                       FORM 10-Q
                                                                       PART I
                                                                       ITEM 1
                                                                       PAGE 2
                               BIOPURE CORPORATION

                 Condensed Consolidated Statements of Operations
                      (In thousands, except per share data)
                                   (Unaudited)

<TABLE>
<CAPTION>
                                                                              Three Months Ended
                                                               -----------------------------------------------
                                                               January 31, 2001               January 29, 2000
                                                               ----------------               ----------------
<S>                                                            <C>                            <C>
Revenues:
         Oxyglobin                                                $       731                    $      595
         Other                                                              4                             3
                                                                  -----------                    ----------
                Total revenues                                            735                           598
Cost of revenues                                                          770                         1,189
                                                                  -----------                    ----------
Gross profit (loss)                                                       (35)                         (591)

Operating expenses:
         Research and development                                       8,187                         8,259
         Sales and marketing                                              622                           579
         General and administrative                                     3,073                         3,112
                                                                  -----------                    ----------
                Total operating expenses                               11,882                        11,950
                                                                  -----------                    ----------

Loss from operations                                                  (11,917)                      (12,541)
Other income, net                                                       1,316                           358
                                                                  -----------                    ----------

Net loss                                                          $   (10,601)                   $  (12,183)
                                                                  ===========                    ==========
Per share data:
         Basic and diluted net loss
         per common share                                         $     (0.42)                   $    (0.55)
                                                                  ===========                    ==========

         Weighted average shares used in
         computing basic net loss
         per common share                                              24,960                        22,282
                                                                  ===========                    ==========
</TABLE>

                             See accompanying notes.

<PAGE>   5

                                                                       FORM 10-Q
                                                                       PART I
                                                                       ITEM 1
                                                                       PAGE 3
                               BIOPURE CORPORATION

                 Condensed Consolidated Statement of Cash Flows
                                 (In thousands)
                                   (Unaudited)

<TABLE>
<CAPTION>
                                                                              Three Months Ended
                                                                  -------------------------------------------
                                                                  January 31, 2001           January 29, 2000
                                                                  ----------------           ----------------
<S>                                                               <C>                        <C>
Operating activities:
    Net loss                                                          $(10,601)                    $(12,183)

Adjustments to reconcile net loss to net
  cash used in operating activities:
    Depreciation                                                           998                          998
    Equity compensation                                                  1,347                        1,688
    Deferred compensation                                                   18                           17
    Accrued interest on stockholders' notes receivable                     (28)                         (46)
    Changes in assets and liabilities:
      Accounts receivable                                                   57                          (15)
      Inventories                                                         (432)                         618
      Other current assets                                                 (28)                          29
      Accounts payable                                                  (1,423)                        (295)
      Accrued expenses                                                    (304)                         645
                                                                     ----------                  ----------
         Net cash used in operating activities                         (10,396)                      (8,544)

Investing activities:
    Purchase of property, plant and equipment                             (934)                         (53)
    Other assets                                                            21                          (13)
                                                                     ---------                   ----------
       Net cash used in investing activities                              (913)                         (66)

Financing activities:
    Net proceeds from sale of common stock                                   -                          (30)
    Payment of notes receivable from stockholders                          337                            -
    Proceeds from exercise of options and warrants                          73                            8
                                                                     ---------                   ----------
            Net cash provided by (used in) financing activities            410                          (22)

Net decrease in cash and cash equivalents                              (10,899)                      (8,632)
Cash and cash equivalents at beginning of period                        88,828                       30,778
                                                                     ---------                   ----------
Cash and cash equivalents at end of period                           $  77,929                   $   22,146
                                                                     =========                   ==========
</TABLE>

                             See accompanying notes.

<PAGE>   6

                                                                       FORM 10-Q
                                                                       PART I
                                                                       ITEM 1
                                                                       PAGE 4

                               BIOPURE CORPORATION

              Notes to Condensed Consolidated Financial Statements
                                January 31, 2001
                                   (Unaudited)

1.   BASIS OF PRESENTATION

          The accompanying unaudited condensed consolidated financial statements
          have been prepared in accordance with accounting principles generally
          accepted in the United States for interim financial information and
          pursuant to the rules and regulations of the Securities and Exchange
          Commission (SEC). Accordingly, they do not include all of the
          information and footnotes required by generally accepted accounting
          principles for complete financial statements.

          In the opinion of management, all adjustments (consisting of normal
          recurring adjustments) considered necessary for a fair presentation
          have been included. Operating results for the three month period ended
          January 31, 2001 are not necessarily indicative of the results that
          may be expected for the year ended October 31, 2001.

          For further information, refer to the consolidated financial
          statements and footnotes thereto for the year ended October 31, 2000,
          included in the Company's Annual Report on Form 10-K for the year
          ended October 31, 2000.

2.   RECENTLY ISSUED ACCOUNTING STANDARDS

          On November 1, 2000, the Company adopted Financial Accounting
          Standards Board Statement (FASB) SFAS No. 133, Accounting for
          Derivative Instruments and Hedging Activities. This statement provides
          a comprehensive and consistent standard for the recognition and
          measurement of derivatives and hedging activities. Adoption of this
          standard had no material effect on the Company's financial position or
          results of operations.

3.   NET LOSS PER SHARE

          Basic net loss per share is computed based on the weighted-average
          number of common shares outstanding during the period. Diluted net
          loss per share is computed based upon the weighted-average number of
          common shares outstanding during the year, adjusted for the dilutive
          effect of shares issuable upon the conversion of preferred stock
          outstanding and the exercise of common stock options and warrants
          determined based upon the average market price of common stock for the
          period. Since the Company has a net loss for all periods presented,
          the effect of all potentially dilutive securities is antidilutive.
          Accordingly, basic and diluted net loss per share are the same.

<PAGE>   7

                                                                       FORM 10-Q
                                                                       PART I
                                                                       ITEM 1
                                                                       PAGE 5
                               BIOPURE CORPORATION

              Notes to Condensed Consolidated Financial Statements
                                January 31, 2001
                                   (Unaudited)
                                   (Continued)

4.   INVENTORIES

          Inventories are valued at the lower of cost (determined using the
          first-in, first-out method) or market. Inventories were as follows:

<TABLE>
<CAPTION>
                                                     January 31, 2001        October 31, 2000
                                                     ----------------        ----------------
<S>                                                  <C>                     <C>
          In thousands
          Raw materials .......................         $    1,150              $    980
          Work-in-process .....................                774                   476
          Finished goods ......................              1,234                 1,270
                                                        ----------              --------
                                                        $    3,158              $  2,726
                                                        ==========              ========
</TABLE>

5.   ACCRUED EXPENSES

          Accrued expenses consisted of the following:

<TABLE>
<CAPTION>
                                                     January 31, 2001        October 31, 2000
                                                     ----------------        ----------------
<S>                                                  <C>                     <C>
          In thousands
          Settlement                                    $    3,508              $  3,508
          Phase III clinical trial                           2,565                 2,897
          BLA                                                  594                     -
          Other                                              2,257                 2,823
                                                        ----------              --------
                                                        $    8,924              $  9,228
                                                        ==========              ========
</TABLE>

6.   STOCK OPTION COMPENSATION

          In August 1999, Biopure granted 386,680 options to two directors which
          are assumed to be for services in addition to board activities and
          must be accounted for at fair value, pursuant to Financial Accounting
          Standards Board Statement No. 123, Accounting for Stock-Based
          Compensation (SFAS 123) and EITF 96-18, Accounting for Equity
          Instruments That Are Issued to Other than Employees for Acquiring, or
          in Conjunction with Selling Goods or Services. The Company records
          compensation expense based on the service period of two years adjusted
          for the fair value of the stock until the options have been earned as
          discussed below. In October 2000 and January 2001, Biopure granted
          30,000 and 7,503 options, respectively, to two consultants, which must
          be accounted for at fair value. With respect to these 424,183 options,
          $1,347,000 was charged to non-cash compensation expense for the
          quarter ended January 31, 2001.

<PAGE>   8

                                                                       FORM 10-Q
                                                                       PART I
                                                                       ITEM 1
                                                                       PAGE 6
                               BIOPURE CORPORATION

              Notes to Condensed Consolidated Financial Statements
                                January 31, 2001
                                   (Unaudited)
                                   (Continued)


          Because the options are assumed to be earned over the service period,
          the final value of the options (i.e. the compensation expense), will
          not be determined until the service period is complete. Therefore, at
          each time interval (i.e. quarterly), the options must be marked to
          fair value (based on a valuation model such as the Black Scholes
          option pricing model). Compensation expense based on fair value will
          continue to be amortized to general and administrative expense on a
          straight-line basis over the vesting period. However, the amount
          charged to expense will be increased or decreased at each point in
          time based on the then current fair value of the Company's stock.

          Therefore, during each time interval the Company will be required to
          record two amounts, the first relating to the current period
          compensation charge and the second would be a cumulative catch-up
          adjustment for the period based on the fair value of the options at
          the end of that period.


7.   COMMITMENT

          On January 31, 2001 the Company signed a letter of intent for the
          construction and financing of an $85,000,000 state-of-the-art
          biopharmaceutical facility in Sumter, South Carolina. Biopure will
          contribute $10,000,000 of equity to the project and enter into a
          20-year lease with optional buyout provisions and a bargain purchase
          option at the end of the term. Lease payments are expected to start in
          the beginning of fiscal 2004. Financing for the new plant will consist
          of asset-backed mortgage financing, tax exempt bonds, equipment lease
          financing, and state and local incentives.


8.   LITIGATION

          The Company was a party to litigation initially filed in 1990 arising
          from certain joint venture agreements for development and distribution
          of product in Central and South America. Summary judgments were
          entered against the two plaintiffs in 1994. The plaintiffs each
          appealed the judgments; one of the appeals was voluntarily dismissed.
          The other appeal was remanded to the trial court for further findings
          based on lack of jurisdiction. The trial court made the requisite
          findings in favor of the Company. The matter was appealed again. By
          judgment entered March 6, 2001, the U.S. Court of Appeals for the
          First Circuit affirmed the summary judgments. In connection with the
          summary judgments, the Company had agreed to a settlement with a
          third-party intervenor with claims against one of the plaintiffs and
          provided for such settlement amount, to be paid in the event of a
          favorable, nonappealable decision by the Court of Appeals, in the
          accompanying financial statements. At January 31, 2001, the Company
          had $3,508,000 in escrow in connection with this settlement and
          included this amount in current portion of restricted cash. The
          settlement amount has been recorded as a current obligation.

<PAGE>   9

                                                                       FORM 10-Q
                                                                       PART I
                                                                       ITEM 2
                                                                       PAGE 7
                               BIOPURE CORPORATION

                     Management's Discussion and Analysis of
                  Financial Condition and Results of Operations
                                January 31, 2001

Except for historical information contained herein, some matters discussed in
this report constitute forward-looking statements within the meaning of Section
27A of the Securities Act of 1933, as amended, and Section 21E of the Securities
Exchange Act of 1934, as amended. Actual results could differ materially from
those projected in the forward-looking statements as a result of the risk
factors set forth in this report and in Exhibit 99.0 to the Company's Annual
Report on Form 10-K for the year ended October 31, 2000. In light of the
substantial risks and uncertainties inherent in all future projections, the
inclusion of forward-looking statements in this report should not be regarded as
representations by the Company that the objectives or plans of the Company will
be achieved. Many factors could cause the Company's actual results, performance
or achievements to differ materially from those in the forward-looking
statements. Reference is made in particular to the discussions set forth below
in this Report under "Management's Discussion and Analysis of Financial
Condition and Results of Operations."

RESULTS OF OPERATIONS

THREE MONTHS ENDED JANUARY 31, 2001 COMPARED TO THREE MONTHS ENDED JANUARY 29,
2000

Total revenues increased 22.9% to $735,000 in the first quarter of fiscal 2001
from $598,000 in the first quarter of fiscal 2000. The increase in revenues was
due to more than a 20% growth in unit sales of Oxyglobin.

Cost of revenues decreased 35.2% to $770,000 in the first quarter of fiscal 2001
from $1,189,000 in the first quarter of fiscal 2000. This decline was primarily
due to increased manufacturing activity associated with the development of
Hemopure, the Company's advanced stage oxygen-carrying drug for human use, and
the resulting decrease in manufacturing expenses allocated to Oxyglobin.

Research and development expenses decreased 0.9% to $8,187,000 in the first
quarter of fiscal 2001 from $8,259,000 in the first quarter of fiscal 2000. In
fiscal 2001, there was a reduction of expenses associated with the pivotal Phase
III clinical trial activities for Hemopure as compared to the same period last
year. This reduction was partially offset by an increase in expenses due to the
preparation for the filing of a U.S. marketing application and ongoing research
and development efforts as compared to the same period last year.

Sales and marketing expenses increased 7.4% to $622,000 in the first quarter of
fiscal 2001 from $579,000 in the first quarter of fiscal 2000. This increase was
primarily due to initial marketing efforts of introducing Oxyglobin in selected
European countries in 2001.

General and administrative expenses decreased 1.3% to $3,073,000 in the first
quarter of fiscal 2001 from $3,112,000 in the first quarter of fiscal 2000. This
decrease was due to non-cash compensation expense for stock options and warrants
granted to certain consultants and directors that must be accounted for at fair
value, per SFAS 123 and EITF 96-18, and be amortized over the service period and
revalued each quarter based on the closing stock price. For the first quarter of
fiscal 2001, we charged $1,347,000 to compensation expense versus $1,687,000 in
the first fiscal quarter of 2000. This reduction of compensation expense was
partially offset by a $255,000 increase in spending on pre-approval marketing
expenses for Hemopure in the first quarter of fiscal 2001 as compared to no
expense for the same period in 2000.

<PAGE>   10

                                                                       FORM 10-Q
                                                                       PART I
                                                                       ITEM 2
                                                                       PAGE 8
                               BIOPURE CORPORATION

                     Management's Discussion and Analysis of
                  Financial Condition and Results of Operations
                                January 31, 2001
                                   (Continued)

Total other income was $1,316,000 in the first quarter of fiscal 2001 compared
to $358,000 in the first quarter of fiscal 2000. Net proceeds from our follow-on
offering of 2,565,000 shares of Class A Common Stock, in March 2000, were
$83,775,000. As a result of the increase in cash, we earned $1,334,000 in
interest during the first quarter of fiscal 2001 versus $374,000 during the
first quarter of fiscal 2000.


LIQUIDITY AND CAPITAL RESOURCES

At January 31, 2001, we had current assets of $85,424,000, consisting primarily
of $77,929,000 in cash and cash equivalents and $3,158,000 in net inventory.
Current assets also included $3,508,000 in current portion of restricted cash.
This amount is to be paid out after the final disposition of litigation, which
had been settled in part for this amount, as described in note 8 to the
financial statements included in this report. Cash included the net proceeds of
$83,775,000 from our follow-on offering.

During the first quarter of fiscal 2001, our cash and cash equivalents decreased
$10,899,000 from $88,828,000 at October 31, 2000 to $77,929,000 at January 31,
2001. This decrease is primarily due to cash outflows of $10,396,000 for
operating activities.

On January 31, 2001 we signed a letter of intent for the construction and
financing of an $85,000,000 state-of-the-art biopharmaceutical facility in
Sumter, South Carolina. Biopure will contribute $10,000,000 of equity to the
project and enter into a 20-year lease with optional buyout provisions and a
bargain purchase option at the end of the term. Lease payments are expected to
start in the beginning of fiscal 2004. Financing for the new plant will consist
of asset-backed mortgage financing, tax exempt bonds, equipment lease financing,
and state and local incentives.

We have financed operations from inception primarily through sales of equity
securities, development and license agreement payments, interest income and
debt.

Our primary investment objective is preservation of principal and we currently
invest in high-grade commercial paper.

We have not been profitable since inception and had an accumulated deficit of
$297,030,000 as of January 31, 2001. We will continue to generate losses from
operations for the foreseeable future. We will explore opportunities to raise
capital through sales of equity and debt securities, potential partnerships,
bank borrowings or leasing arrangements.

We believe our current cash, cash equivalents and short-term investments will be
sufficient to meet our projected operating requirements through the first half
of fiscal 2002. Our cash requirements may vary significantly from current
projections.

As of October 31, 2000, we had net operating loss carryforwards of approximately
$171,000,000 to offset future federal and state taxable income through 2020. Due
to the degree of uncertainty related to the ultimate realization of such prior
losses, no benefit has been recognized in our financial statements as of October
31, 2000. Utilization of such losses in future years may be limited under the
change of stock ownership rules of the Internal Revenue Service.

<PAGE>   11

                                                                       FORM 10-Q
                                                                       PART 1
                                                                       ITEM 3
                                                                       PAGE 9
                               BIOPURE CORPORATION

                     Management's Discussion and Analysis of
                  Financial Condition and Results of Operations
                                January 31, 2001
                                   (Continued)



QUANTITATIVE AND QUALITATIVE DISCLOSURE OF MARKET RISK

The Company currently does not have any foreign currency exchange risks with the
exception of negligible exchange fluctuations associated with expenses for
clinical trial and regulatory activities outside of the United States. The
Company invests its cash and cash equivalents in high-grade commercial paper and
money market funds. These investments are subject to interest rate risk.
However, due to the nature of the Company's short-term investments, it believes
that the financial market risk exposure is not material.

<PAGE>   12

                                                                       FORM 10-Q
                                                                       PART II
                                                                       ITEM 1-6
                                                                       PAGE 10

                               BIOPURE CORPORATION
                           Part II - Other Information
                                January 31, 2001



ITEM 1 - LEGAL PROCEEDINGS

Reference is made to Item 103 in the Corporation's Annual Report on Form 10-K
for the year ended October 31, 2000. By judgment entered March 6, 2001, the
U.S. Court of Appeals for the First Circuit affirmed the summary judgments in
favor of the Company.

ITEM 2 - CHANGES IN SECURITIES AND USE OF PROCEEDS

Warrants to purchase 25,854 shares of Class A Common Stock were exercised in the
first quarter of fiscal 2001 for aggregate proceeds to the Corporation of
$19,980. The Corporation relied on Section 4(2) of the Securities Act of 1933
and Regulation D under the Securities Act of 1933 in issuing shares upon the
exercise of warrants.


ITEM 6 - EXHIBITS AND REPORTS ON FORM 8-K

     (a)  No exhibits are being filed with this report.
     (b)  No reports on Form 8-K were filed during the quarter for which this
          report is filed.

<PAGE>   13

                                                                       FORM 10-Q
                                                                       PART II
                                                                       PAGE 11


                                   SIGNATURES


Pursuant to the requirements of the Securities Exchange Act of 1934, the
registrant has duly caused this report to be signed on its behalf by the
undersigned thereunto duly authorized.


                                   BIOPURE CORPORATION



     Date:  March 19, 2001         By:  /s/ Carl W. Rausch
                                       -----------------------------------------
                                        CARL W. RAUSCH
                                        Chairman of the Board
                                        Chief Executive Officer


     Date:  March 19, 2001         By:  /s/ Francis H. Murphy
                                       -----------------------------------------
                                        FRANCIS H. MURPHY
                                        Chief Financial Officer