<PAGE>   1
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                                    FORM 10-Q

(Mark One)

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

For the quarterly period ended         March 31, 1998
                               -------------------------------------------------

                                       OR

[ ]  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 0-23378
                                                -------

                         Thermadyne Holdings Corporation
--------------------------------------------------------------------------------
             (Exact Name of Registrant as Specified in Its Charter)


           Delaware                                           74-2482571
--------------------------------------------------------------------------------
(State or Other Jurisdiction of                           (I.R.S. Employer 
Incorporation or Organization)                           Identification No.)

101 S. Hanley, St. Louis, MO                                    63105
--------------------------------------------------------------------------------
(Address of Principal Executive Offices)                      (Zip Code)


Registrant's Telephone Number, Including Area Code       314/721-5573
                                                   -----------------------------

     Indicate by X 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 [ ]

     Indicate by check mark whether the registrant has filed all documents and
reports required to be filed by Section 12, 13 or 15 (d) of the Securities
Exchange Act of 1934 subsequent to the distribution of securities under a plan
confirmed by a court.              Yes [x]   No [ ]

     The number of shares outstanding of the issuer's common stock, par value
$0.01 per share, as of May 1, 1998 was 11,217,233.


<PAGE>   2

                         THERMADYNE HOLDINGS CORPORATION

                                      INDEX


PART I - FINANCIAL INFORMATION

         Item 1.  Financial Statements

         Consolidated Balance Sheets ......................................... 3
         Consolidated Statements of Income ................................... 4
         Consolidated Statements of Cash Flows ............................... 5
         Notes to Consolidated Financial Statements ........................ 6-8

         Item 2.  Management's Discussion and Analysis
                  of Financial Condition and Results of Operations .........9-10


PART II - OTHER INFORMATION

         Item 6.  Exhibits and Reports on Form 8-K .......................... 11


SIGNATURES .................................................................. 12



<PAGE>   3

                         THERMADYNE HOLDINGS CORPORATION
                           CONSOLIDATED BALANCE SHEETS
                        (In thousands, except share data)

<TABLE>
<CAPTION>
                                                                         March 31,     December 31,
                                                                           1998            1997
                                                                         ---------      ---------
                                                                        (Unaudited)
ASSETS
<S>                                                                      <C>            <C>      
Current assets:
         Cash and cash equivalents                                       $     173      $   1,481
         Accounts receivable, less allowance for
           doubtful accounts of $2,278 and
           $2,217, respectively                                             82,171         76,847
         Inventories                                                       118,966        105,135
         Prepaid expenses and other                                          8,316          8,534
                                                                         ---------      ---------
           Total current assets                                            209,626        191,997
Property, plant and equipment, at cost, net                                 86,025         85,257
Deferred financing costs, net                                                5,550          5,754
Intangibles, at cost, net                                                   33,797         33,970
Deferred income taxes                                                       35,551         35,552
Other assets                                                                 1,893          1,997
                                                                         ---------      ---------
           Total assets                                                  $ 372,442      $ 354,527
                                                                         =========      =========

LIABILITIES AND SHAREHOLDERS' DEFICIT

Current liabilities:
         Accounts payable                                                $  49,906      $  55,390
         Accrued and other liabilities                                      26,478         32,697
         Accrued interest                                                   12,976          5,680
         Income taxes payable                                                9,725          4,769
         Current maturities of long-term obligations                         6,256          4,912
                                                                         ---------      ---------
           Total current liabilities                                       105,341        103,448
Long-term obligations, less current maturities                             364,160        353,175
Other long-term liabilities                                                 59,718         60,751
Shareholders' equity (deficit):
         Common stock, $.01 par value, 25,000,000
           shares authorized, 11,217,233 and 11,189,675 shares
           issued and outstanding at March 31, 1998 and
           December 31, 1997, respectively                                     112            112
         Additional paid-in capital                                        149,358        149,023
         Accumulated deficit                                              (293,399)      (299,208)
         Foreign currency translation                                      (12,848)       (12,774)
                                                                         ---------      ---------
           Total shareholders' deficit                                    (156,777)      (162,847)
                                                                         ---------      ---------
           Total liabilities and shareholders' deficit                   $ 372,442      $ 354,527
                                                                         =========      =========
</TABLE>


           See accompanying notes to consolidated financial statements



                                       3
<PAGE>   4


                         THERMADYNE HOLDINGS CORPORATION
                        CONSOLIDATED STATEMENTS OF INCOME
                        (In thousands, except share data)
                                   (Unaudited)


<TABLE>
<CAPTION>
                                                                    Three Months    Three Months
                                                                       Ended           Ended
                                                                   March 31, 1998  March 31, 1997  
                                                                   --------------  --------------  
<S>                                                                 <C>            <C>      
Net sales                                                           $ 131,829      $ 117,751
Operating expenses:
         Cost of goods sold                                            81,784         70,342
         Selling, general and administrative expenses                  27,064         26,270
         Amortization of goodwill                                         382            357
         Amortization of other intangibles                                518          1,692
         Net periodic postretirement benefits                             650            585
                                                                    ---------      ---------
Operating income                                                       21,431         18,505
Other expense:
         Interest expense                                             (10,834)       (11,538)
         Amortization of deferred financing costs                        (370)          (460)
         Other, net                                                       166            406
                                                                    ---------      ---------
Income from continuing operations before income tax
         provision                                                     10,393          6,913
Income tax provision                                                    4,584          2,981
                                                                    ---------      ---------
Income from continuing operations                                       5,809          3,932
Discontinued operations:
         Gain from discontinued operations, net of income taxes            --          1,036
                                                                    ---------      ---------
Net income                                                          $   5,809      $   4,968
                                                                    =========      =========

Basic earnings per share amounts:
         Income from continuing operations                          $    0.52      $    0.36
         Net income                                                      0.52           0.45

Diluted earnings per share amounts:
         Income from continuing operations                               0.50           0.35
         Net income                                                      0.50           0.44
</TABLE>


          See accompanying notes to consolidated financial statements




                                       4
<PAGE>   5
                         THERMADYNE HOLDINGS CORPORATION
                      CONSOLIDATED STATEMENTS OF CASH FLOWS
                                 (In thousands)
                                   (Unaudited)


<TABLE>
<CAPTION>
                                                                          Three Months     Three Months
                                                                              Ended            Ended
                                                                          March 31, 1998   March 31, 1997
                                                                          --------------   --------------
<S>                                                                        <C>               <C>
Cash flows provided by (used in) operating activities: 
     Net income                                                            $  5,809          $  4,968  
     Adjustments to reconcile net income to net cash provided by (used                                 
     in) operating activities:                                                                         
        Net periodic postretirement benefits                                    650               585  
        Depreciation                                                          3,556             2,876  
        Amortization of goodwill                                                382               357  
        Amortization of other intangibles                                       518             1,692  
        Amortization of deferred financing costs                                370               460  
        Recognition of net operating loss carryforwards                        --                 586  
        Deferred income taxes                                                  --                (586) 
        Noncash charges for discontinued operations                            --                 565  
Changes in operating assets and liabilities:                                                           
        Accounts receivable                                                  (6,179)           (2,577) 
        Inventories                                                         (13,502)           (4,625) 
        Prepaid expenses and other                                              (52)             (334) 
        Accounts payable                                                     (5,347)              358  
        Accrued and other liabilities                                        (6,246)           (7,762) 
        Accrued interest                                                      7,370             7,249  
        Income taxes payable                                                  5,098            (1,513) 
        Other long-term liabilities                                          (1,562)             (321) 
        Discontinued operations                                                --                 903  
                                                                           --------          --------  
          Total adjustments                                                 (14,944)           (2,087) 
                                                                           --------          --------  
              Net cash provided by (used in) operating activities            (9,135)            2,881  
                                                                           --------          --------  
Cash flows provided by (used in) investing activities:                                                 
        Capital expenditures, net                                            (3,756)           (2,395) 
        Change in other assets                                                 (349)            8,512  
        Acquisitions, net of cash                                              (640)          (27,755) 
        Investing activities of discontinued operations                        --                (570) 
                                                                           --------          --------  
              Net cash used in investing activities                          (4,745)          (22,208) 
                                                                           --------          --------  
Cash flows provided by (used in) financing activities:                                                 
        Change in long-term receivables                                         263                17  
        Repayment of long-term obligations                                  (10,592)           (9,928) 
        Borrowing of long-term obligations                                   22,370            33,683  
        Issuance of common stock                                                335               436  
        Change in accounts receivable securitization                            376             4,631  
        Financing activities of discontinued operations                          --            (1,227)
        Other                                                                  (180)           (2,168) 
                                                                           --------          --------  
              Net cash provided by financing activities                      12,572            25,444  
                                                                           --------          --------  
Net increase (decrease) in cash and cash equivalents                         (1,308)            6,117  
Cash and cash equivalents at beginning of period                              1,481             1,420  
                                                                           --------          --------  
Cash and cash equivalents at end of period                                 $    173          $  7,537  
                                                                           ========          ========  
</TABLE>


         See accompanying notes to consolidated financial statements


                                       5
<PAGE>   6




                         THERMADYNE HOLDINGS CORPORATION
                   NOTES TO CONSOLIDATED FINANCIAL STATEMENTS
                                 (In thousands)
                                   (Unaudited)


1.       BASIS OF PRESENTATION

         UNAUDITED INTERIM CONSOLIDATED FINANCIAL STATEMENTS

         The unaudited interim consolidated financial statements reflect all
material adjustments (only of a normal recurring nature) which are, in the
opinion of management, necessary for a fair presentation of financial position
and results of operations. The results for the three months ended March 31, 1998
are not necessarily indicative of the results that may be expected for a full
fiscal year.

         STATEMENTS OF CASH FLOWS

         For purposes of the Statements of Cash Flows, the Company considers all
highly liquid investments purchased with a maturity of three months or less to
be cash equivalents. Interest and taxes paid were as follows:


<TABLE>
<CAPTION>
                       Three Months     Three Months  
                          Ended             Ended     
                       March 31, 1998   March 31, 1997
                       --------------   --------------
<S>                    <C>              <C>       
Interest               $ 3,538             $ 4,561   
Taxes                     (624)              3,801   
</TABLE>


         EARNINGS PER SHARE

         In February 1997, the Financial Accounting Standards Board issued
Statement No. 128, Earnings per Share ("FASB 128"). FASB 128 replaced the
calculation of primary and fully diluted earnings per share with basic and
diluted earnings per share. Unlike primary earnings per share, basic earnings
per share excludes any dilutive effects of options, warrants and convertible
securities. Diluted earnings per share is very similar to the previously
reported fully diluted earnings per share. All earning per share amounts for all
periods have been presented to conform to the Statement 128 requirements. All
exchange arrangements contemplated by the Company's 1994 financial restructuring
are assumed to have been completed.



                                       6
<PAGE>   7

<TABLE>
<CAPTION>
                                                            Three Months      Three Months
                                                               Ended              Ended
                                                            March 31, 1998    March 31, 1997
<S>                                                          <C>                <C>       
Basic earnings per share amounts:
Income from continuing operations                            $     0.52         $     0.36
Discontinued operations                                             --                0.09
                                                             ----------         ----------
Net income                                                   $     0.52         $     0.45
                                                             ==========         ==========

Diluted earnings per share amounts:
Income from continuing operations                            $     0.50         $     0.35
Discontinued operations                                             --                0.09
                                                             ----------         ----------
Net income                                                   $     0.50         $     0.44
                                                             ==========         ==========

Weighted average shares - basic earnings per share           11,208,536         11,028,126
Effect of dilutive securities:
   Employee stock options                                       328,064            287,571
                                                             ----------         ----------
Weighted average shares - diluted earnings per share         11,536,600         11,315,697
                                                             ==========         ==========
</TABLE>


2.       INVENTORIES

         The composition of inventories at March 31, 1998 was as follows:


<TABLE>
<S>                                           <C>      
                          Raw materials       $  20,612
                          Work-in-process        25,874
                          Finished goods         73,240
                          LIFO Reserve             (760)
                                                       
                                              ---------
                                Total         $ 118,966
                                              =========

</TABLE>


3.       MERGER WITH MERCURY ACQUISITION CORPORATION

         On January 20, 1998, the Company and Mercury Acquisition Corporation
("Mercury") an affiliate of DLJ Merchant Banking Partners II, L.P. ("DLJ"),
entered into a definitive merger agreement (the "Merger Agreement"). Under the
terms of the Merger Agreement, Mercury will merge with and into the Company,
and, subject to the following sentence, the holders of each share of the
Company's common stock can elect to receive $34.50 in cash for such share or to
retain such share in the merged Company. In any event, holders will be required
to retain 485,010 shares, or approximately 4.3%, of the Company's common stock
outstanding immediately prior to the merger. In addition, DLJ has entered into
voting agreements with Magten Asset Management (on behalf of itself and certain
of its clients) and Fidelity Capital and Income Fund, pursuant to which these
current shareholders, subject to certain conditions, have 


                                       7
<PAGE>   8

agreed to vote in favor of the merger 5,942,708 shares of the Company's common
stock owned by them. These shares represent approximately 53% of the Company's
common stock outstanding on December 31, 1997.

         That proposed merger, which will be recorded as a recapitalization for
accounting purposes, is subject to a number of conditions, including regulatory
approvals and approval by Company shareholders. The transaction is estimated to
have an aggregate value of approximately $790 million, including refinancing of
the Company's existing revolving credit facility and senior and subordinated
notes. The Company expects the merger to close by June 30, 1998.

         As a result of the proposed merger, the Company and Mercury will incur
various costs, currently estimated to range between $50 million and $60 million,
on a pretax basis, in connection with consummating the transaction. These costs
consist primarily of professional fees, registration costs, compensation costs
and other expenses. Although the exact timing, nature and amount of these merger
transaction costs are subject to change, the Company expects that a one-time
charge for these costs will be recorded in the quarter during which the merger
in consummated.

4.       RECENT ACCOUNTING PRONOUNCEMENT

         As of January 1, 1998, the Company adopted Statement 130, Reporting
Comprehensive Income. Statement 130 establishes new rules for the reporting and
display of comprehensive income and its components; however, the adoption of
this Statement had no impact on the Company's net income or shareholders'
equity. Statement 130 requires foreign currency translation adjustments, which
prior to adoption were reported separately in shareholders' equity to be
included in other comprehensive income. Prior year financial statements have
been reclassified to conform to the requirements of Statement 130.

         During the first quarter of 1998 and 1997, total comprehensive income
amounted to $5,735 and $1,724.



                                       8
<PAGE>   9


                     MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                  FINANCIAL CONDITION AND RESULTS OF OPERATIONS

         Included in the following discussions are comparisons of Adjusted
EBITDA which is defined as operating income plus depreciation, amortization of
goodwill, amortization of intangibles and net periodic postretirement benefits
expense and is a key financial measure but should not be construed as an
alternative to operating income or cash flows from operating activities (as
determined in accordance with generally accepted accounting principles).
Adjusted EBITDA is also one of the financial measures by which the Company's
compliance with its covenants is calculated under its debt agreements. The
Company believes that Adjusted EBITDA is a useful supplement to net income
(loss) and other consolidated income statement data in understanding cash flows
generated from operations that are available for taxes, debt service and
capital expenditures. However, the Company's method of computation may or may
not be comparable to other similarly titled measures of other companies. In
addition, Adjusted EBITDA is not necessarily indicative of amounts that may be
available for discretionary uses and does not reflect any legal or contractual
restrictions on the Company's use of funds.

RESULTS OF OPERATIONS

Three Months ended March 31, 1998 compared to Three Months ended March 31, 1997

         Net sales were $131.8 million for the three months ended March 31, 1998
compared to $117.8 million for the three months ended March 31, 1997, an
increase of 12.0%. Domestic sales for the first three months of 1998 were up
20.3% (10.5% excluding acquisitions) over the same period of a year ago.
International sales increased approximately 1.0% overall despite the continuing
economic turmoil in Asia. Europe, Latin America and the Middle East regions all
recorded significant increases over the prior year.

         Cost of goods sold as a percentage of sales for the quarter ended March
31, 1998 was 62.0% compared to 59.7% for the quarter ended March 31, 1997. This
increase is the result of recent acquisitions which generally carry a somewhat
lower average gross margin than the Company's existing businesses, and sales mix
as the Company expands its product offering to include items in more
price-competitive markets.

         Selling, general and administrative expenses were $27.1 million for the
first three months of 1998, an increase of $0.8 million, or 3.0% over the first
three months of 1997. As a percentage of sales, selling, general and
administrative expenses were 20.5% for the three months ended March 31, 1998 and
22.3% for the three months ended March 31, 1997. This improvement is due in part
to the Company's growth in sales as a portion of its costs are fixed in nature
as well as the realization of savings from a cost reduction program the Company
initiated in December 1997.


                                       9
<PAGE>   10

         Interest expense decreased from $11.5 million in the first quarter of
1997 to $10.8 million in the first quarter of 1998, a decrease of $0.7 million
or 6.1%. This is largely the result of lower debt levels in 1998 compared to
1997.

         Income tax provision was $4.6 million on pre-tax income of $10.4
million for the three months ended March 31, 1998, compared to an income tax
provision of $3.0 million on pre-tax income of $6.9 million for the three months
ended March 31, 1997.

         Adjusted EBITDA was $26.5 million for the quarter ended March 31, 1998
compared to $24.0 million for the quarter ended March 31, 1997, an increase of
$2.5 million or 10.5%.

LIQUIDITY AND CAPITAL RESOURCES

         Working Capital and Cash Flows. Cash used in operating activities was
$9.1 million for the first quarter of 1998, a decrease of $12.0 million from the
first quarter of 1997 in which cash flow from operations provided $2.9 million.
This use of cash by operating activities is the result of a net increase in
operating assets and liabilities of $12.9 million. Net cash used in investing
activities decreased from $22.2 million in the first quarter of 1997 to $4.7
million in the first quarter of 1998. Cash used for acquisitions decreased $27.1
million and was offset by an increase in capital expenditures of $1.4 million
and a decrease in cash provided by other assets of $8.9 million. Cash provided
by financing activities decreased $12.9 million to $12.6 million for the quarter
ended March 31, 1998 from $25.4 million for the quarter ended March 31, 1997.
Net borrowings of long-term debt decreased $12.0 million in a comparison of
these same periods. In addition, the accounts receivable securitization program
provided $4.3 million less in the first three months of 1998 compared to 1997,
while financing activities of discontinued operations and other financing
activities used $1.2 million and $2.0 million less in the first quarter of 1998,
respectively.

         Liquidity. The major uses of cash in the remainder of 1998 are expected
to be for debt service requirements, capital expenditures and tax payments.
Management believes that cash from operating activities, together with available
borrowings under its revolving credit facility, if necessary, will be sufficient
to permit the Company to meet these financial obligations.

         The Company will continue from time to time to explore additional
auxiliary financing methods and other means to lower its cost of capital, which
could include stock issuances or debt financing and the application of the
proceeds therefrom to the payment of bank debt, or the purchase of senior or
senior subordinated notes.


                                       10
<PAGE>   11


PART II.  OTHER INFORMATION

Item 6.  Exhibits and Reports on Form 8-K

         a)       Exhibits:
                  27 - Financial Data Schedule

         b)       Reports on Form 8-K.

                  A Current Report on Form 8-K dated January 21, 1998 was filed
by the Company regarding its merger with an affiliate of DLJ Merchant Banking
Partners II.

                  A Current Report on Form 8-K/A dated February 23, 1998 was
filed by the Company regarding its merger with an affiliate of DLJ Merchant
Banking Partners II.



                                       11
<PAGE>   12




                                   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.


                                        THERMADYNE HOLDINGS CORPORATION


                                        By:  /s/  RANDALL E. CURRAN
                                           -------------------------------------
                                            Randall E. Curran
                                            Chairman of the Board, President and
                                            Chief Executive Officer
                                            (Principal Executive Officer)




                                        By:  /s/  JAMES H. TATE
                                           -------------------------------------
                                            James H. Tate
                                            Senior Vice President &
                                            Chief Financial Officer
                                            (Principal Financial and 
                                             Accounting Officer)






Date:  May 15, 1998
     --------------




                                       12
<PAGE>   13


                              INDEX TO EXHIBITS




<TABLE>
<CAPTION>
EXHIBIT
NUMBER              DESCRIPTION
------              -----------
<S>                 <C>
 27                 Financial Data Schedule
</TABLE>