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


                                  UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                                    FORM 10-Q

             [X] QUARTERLY REPORT UNDER 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 000-22433

                           BRIGHAM EXPLORATION COMPANY
             (Exact name of registrant as specified in its charter)

           DELAWARE                        1311                 75-2692967
(State of other jurisdiction   (Primary Standard Industrial  (I.R.S. Employer
      of incorporation                Classification          Identification 
      or organization)                  Code Number)                Number)

                            6300 BRIDGE POINT PARKWAY
                               BLDG. 2, SUITE 500
                               AUSTIN, TEXAS 78730
                                 (512) 427-3300

            (Name, address, including zip code, and telephone number,
       including area code, of Registrant's principal executive offices)

   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 twelve months (or 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 [ ].

As of April 30, 1998, 12,253,574 shares of Common Stock, $.01 per share, were
outstanding.

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

<PAGE>   2

                           BRIGHAM EXPLORATION COMPANY

                                      INDEX

<TABLE>
<CAPTION>
                                                                                               PAGE
PART I.       FINANCIAL INFORMATION:                                                          NUMBER
                                                                                              ------
<S>           <C>                                                                             <C>
Item 1.       Unaudited Condensed Consolidated Financial Statements

              a)   Balance Sheets - December 31, 1997 and
                      March 31, 1998                                                             1

              b)   Statements of Operations - Three months ended
                      March 31, 1997 and 1998                                                    2

              c)   Statements of Cash Flows - Three months ended
                      March 31, 1997 and 1998                                                    3

              d)   Notes to Consolidated Financial Statements                                  4 - 6

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


PART II.      OTHER INFORMATION:

Item 6.       Exhibits and Reports on Form 8-K                                                9 - 10
</TABLE>


<PAGE>   3
PART I.  FINANCIAL INFORMATION:

Item 1.  Financial Statements

                           BRIGHAM EXPLORATION COMPANY

                             CONDENSED CONSOLIDATED
                                 BALANCE SHEETS
                                 (in thousands)

<TABLE>
<CAPTION>
                                                                                         December 31,      March 31,
                                                                                             1997            1998
                                                                                          ----------      ----------
                                                                                                          (unaudited)
<S>                                                                                       <C>             <C>       
                                                         ASSETS
Current assets:
     Cash and cash equivalents                                                            $    1,701      $    1,800
     Accounts receivable                                                                       4,909           5,510
     Prepaid expenses                                                                            280             267
                                                                                          ----------      ----------
         Total current assets                                                                  6,890           7,577
                                                                                          ----------      ----------

Natural gas and oil properties, at cost, net                                                  84,176          96,676
Other property and equipment, at cost, net                                                     1,239           1,374
Drilling advances paid                                                                            78              62
Deferred loan fees                                                                                --           1,805
Other noncurrent assets                                                                           18              14
                                                                                          ----------      ----------
                                                                                          $   92,401      $  107,508
                                                                                          ==========      ==========

                                          LIABILITIES AND STOCKHOLDERS' EQUITY
Current liabilities:
     Accounts payable                                                                     $   11,892      $    5,819
     Accrued drilling costs                                                                    2,406           3,071
     Participant advances received                                                               489           3,110
     Other current liabilities                                                                   726           1,374
                                                                                          ----------      ----------
         Total current liabilities                                                            15,513          13,374
                                                                                          ----------      ----------

Notes payable                                                                                 32,000          50,000
Other noncurrent liabilities                                                                     507             478
Deferred income tax liability                                                                  1,228             914

Stockholders' equity:
     Preferred stock, $.01 par value, 10 million shares
         authorized, none issued and outstanding                                                  --              --
     Common stock, $.01 par value, 30 million shares
         authorized, 12,253,574 issued and outstanding                                           123             123
     Additional paid-in capital                                                               44,344          44,344
     Unearned stock compensation                                                              (1,340)         (1,136)
     Retained earnings (accumulated deficit)                                                      26            (589)
                                                                                          ----------      ----------
         Total stockholders' equity                                                           43,153          42,742
                                                                                          ----------      ----------
                                                                                          $   92,401      $  107,508
                                                                                          ==========      ==========
</TABLE>


   See accompanying notes to the condensed consolidated financial statements.

                                        1
<PAGE>   4

                           BRIGHAM EXPLORATION COMPANY

                             CONDENSED CONSOLIDATED
                            STATEMENTS OF OPERATIONS
                      (in thousands, except per share data)
                                   (unaudited)

<TABLE>
<CAPTION>
                                                                  Three Months
                                                                 Ended March 31,
                                                            --------------------------
                                                               1997            1998
                                                            ----------      ----------
<S>                                                         <C>             <C>       
Revenues:
     Natural gas and oil sales                              $    2,136      $    3,143
     Workstation revenue                                           164             114
                                                            ----------      ----------
                                                                 2,300           3,257
                                                            ----------      ----------
Costs and expenses:
     Lease operating                                               206             414
     Production taxes                                              127             188
     General and administrative                                    702           1,154
     Depletion of natural gas and oil properties                   687           1,267
     Depreciation and amortization                                 108              83
     Amortization of stock compensation                             29              95
                                                            ----------      ----------
                                                                 1,859           3,201
                                                            ----------      ----------
         Operating income                                          441              56
                                                            ----------      ----------

Other income (expense):
     Interest income                                                18              37
     Interest expense                                             (216)         (1,022)
     Interest expense - related party                             (174)             --
                                                            ----------      ----------
                                                                  (372)           (985)
                                                            ----------      ----------

Net income (loss) before income taxes                               69            (929)
Income tax (expense) benefit                                    (4,964)            314
                                                            ----------      ----------
     Net loss                                               $   (4,895)     $     (615)
                                                            ==========      ==========

Net loss per share:
     Basic / Diluted                                        $    (0.55)     $    (0.05)

Weighted average common shares outstanding:
     Basic / Diluted                                             8,929          12,254
</TABLE>



   See accompanying notes to the condensed consolidated financial statements.

                                        2
<PAGE>   5

                           BRIGHAM EXPLORATION COMPANY

                             CONDENSED CONSOLIDATED
                            STATEMENTS OF CASH FLOWS
                                 (in thousands)
                                   (unaudited)

<TABLE>
<CAPTION>
                                                                             Three           Three
                                                                          Months Ended    Months Ended
                                                                            March 31,       March 31,
                                                                              1997            1998
                                                                           ----------      ----------
<S>                                                                        <C>             <C>        
Cash flows from operating activities:
     Net loss                                                              $   (4,895)     $     (615)
     Adjustments to reconcile net loss to cash
      provided by operating activities:
         Depletion of natural gas and oil properties                              687           1,267
         Depreciation and amortization                                            108              83
         Amortization of stock compensation                                        29              95
         Amortization of deferred loan fees                                        --             106
         Changes in deferred income tax liability                               4,964            (314)
         Changes in working capital and other items                             3,331          (3,418)
                                                                           ----------      ----------
            Net cash provided (used) by operating activities                    4,224          (2,796)
                                                                           ----------      ----------

Cash flows from investing activities:
     Additions to natural gas and oil properties                               (6,830)        (12,993)
     Additions to other property and equipment                                    (33)           (159)
     (Increase) decrease in drilling advanaces paid                              (321)             16
     Increase in exploration advances paid                                       (117)             --
                                                                           ----------      ----------
            Net cash used by investing activities                              (7,301)        (13,136)
                                                                           ----------      ----------

Cash flows from financing activities:
     Increase in notes payable                                                  2,850          52,800
     Repayment of notes payable                                                    --         (34,800)
     Principal payments on capital lease obligations                              (56)            (58)
     Deferred loan fees                                                            --          (1,911)
                                                                           ----------      ----------
            Net cash provided by financing activities                           2,794          16,031
                                                                           ----------      ----------

Net (decrease) increase in cash and cash equivalents                             (283)             99

Cash and cash equivalents, beginning of period                                  1,447           1,701
                                                                           ----------      ----------
Cash and cash equivalents, end of period                                   $    1,164      $    1,800
                                                                           ==========      ==========
</TABLE>



   See accompanying notes to the condensed consolidated financial statements.

                                        3
<PAGE>   6
                           BRIGHAM EXPLORATION COMPANY

                             NOTES TO THE CONDENSED
                        CONSOLIDATED FINANCIAL STATEMENTS
                                   (UNAUDITED)

1.     ORGANIZATION AND NATURE OF OPERATIONS

       Brigham Exploration Company (the "Company") is a Delaware corporation
       formed on February 25, 1997 for the purpose of exchanging its common
       stock for the common stock of Brigham, Inc. and the partnership interests
       of Brigham Oil & Gas, L.P. (the "Partnership"). Brigham, Inc. is a Texas
       corporation whose only asset is its ownership interest in the
       Partnership. The Partnership was formed in May 1992 to explore and
       develop onshore domestic natural gas and oil properties using 3-D seismic
       imaging and other advanced technologies. Since its inception, the
       Partnership has focused its exploration and development of natural gas
       and oil properties in West Texas, the Anadarko Basin and the onshore Gulf
       Coast.

       Pursuant to an exchange agreement dated February 26, 1997 (the "Exchange
       Agreement") and upon the initial filing on February 27, 1997 of a
       registration statement with the Securities and Exchange Commission for
       the public offering of common stock (the "Offering"), the shareholders of
       Brigham, Inc. transferred all of the outstanding stock of Brigham, Inc.
       to the Company in exchange for 3,859,821 shares of common stock of the
       Company. Pursuant to the Exchange Agreement, the Partnership's other
       general partner and the limited partners also transferred all of their
       partnership interests to the Company in exchange for 3,314,286 shares of
       common stock of the Company. Furthermore, the holders of the
       Partnership's subordinated convertible notes transferred these notes to
       the Company in exchange for 1,754,464 shares of common stock. These
       transactions are referred to as the "Exchange." In completing the
       Exchange, the Company issued 8,928,571 shares of common stock to the
       stockholders of Brigham, Inc., the partners of the Partnership and the
       holder of the Partnership's subordinated notes payable. As a result of
       the Exchange, the Company now owns all the partnership interests in the
       Partnership.

       In May 1997, the Company sold 3,325,000 shares of its common stock in the
       Offering at a price of $8.00 per share. With a portion of the proceeds
       from the Offering, the Company repaid the then outstanding borrowings
       ($13.3 million) under the Company's revolving credit facility.

2.     BASIS OF PRESENTATION

       The unaudited condensed consolidated balance sheets at December 31, 1997
       and March 31, 1998 reflect the consolidated accounts of the Company. The
       unaudited condensed consolidated statements of operations and of cash
       flows for the three months ended March 31, 1997 and 1998 include the
       results of operations and of cash flows of the Partnership for the period
       from January 1, 1997 to February 27, 1997 and of the Company for the
       period from February 25, 1997, the date of its inception, to March 31,
       1997 and for the three months ended March 31, 1998. As the Exchange was
       the conversion of a partnership to a corporation, the Exchange was
       accounted for by the Company as a reorganization.

       The accompanying condensed consolidated financial statements are
       unaudited, and in the opinion of management, reflect all adjustments that
       are necessary for a fair presentation of the financial position and
       results of operations for the periods presented. All such adjustments are
       of a normal and recurring nature. The results of operations for the
       periods presented are not necessarily indicative of the results to be
       expected for the entire year. The unaudited condensed consolidated
       financial statements should be read in conjunction with the Company's
       1997 Annual Report on Form 10-K pursuant to Section 13 or 15(d) of the
       Securities Exchange Act of 1934.



                                        4
<PAGE>   7

                           BRIGHAM EXPLORATION COMPANY

                             NOTES TO THE CONDENSED
                        CONSOLIDATED FINANCIAL STATEMENTS
                                   (UNAUDITED)


3.     NOTES PAYABLE

       In January 1998, the Company entered into a new reserve based revolving
       credit facility (the "Credit Facility"). The Credit Facility provides for
       borrowings up to $75 million, all of which is immediately available for
       borrowing to fund capital expenditures, until January 31, 1999, at which
       time the borrowing availability will be redetermined by the lender based
       on the Company's proved reserve value at that time. The Company may
       elect, at its option, to have the borrowing availability redetermined
       based on the Company's proved reserve value at any time prior to January
       31, 1999. Amounts outstanding under the Credit Facility bear interest at
       either the lender's Base Rate or LIBOR plus 2.25%, at the Company's
       option. The Company's obligations under the Credit Facility are secured
       by substantially all of the natural gas and oil properties of the
       Company. A portion of the funds borrowed under the Credit Facility were
       used to repay in full the debt outstanding under the Company's previous
       revolving credit facility.

       In connection with the origination of the Credit Facility, certain bank
       fees and other expenses totaling approximately $1.9 million were recorded
       as deferred costs and will be amortized over the life of the loan which
       matures January 26, 2001.

4.     INCOME TAXES

       Prior to the consummation of the Exchange, the Partnership was not
       subject to federal income taxes. Income and losses were passed through to
       its partners on the basis of the allocation provisions established by the
       partnership agreement. Upon consummation of the Exchange, the
       Partnership's net income became subject to federal income taxes through
       its ownership by the Company. Also, in conjunction with the Exchange, the
       Company recorded a deferred income tax liability of $5 million to
       recognize the temporary differences between the financial statement and
       tax bases of the assets and liabilities of the Partnership at the
       Exchange date, February 27, 1997, given the provisions of enacted tax
       laws. Subsequent to this date, the Company elected to record a step-up in
       basis of its assets for tax purposes as a result of the Exchange. As a
       result of this election, the Company recorded a $3.8 million deferred
       income tax benefit in the fourth quarter of 1997, which resulted in a net
       $1.2 million non-cash deferred income tax charge for the year ended
       December 31, 1997.

5.     EARNINGS PER SHARE

       Earnings per share have been calculated in accordance with the provisions
       of Statement of Financial Accounting Standards ("SFAS") No. 128. The
       implementation of this standard has resulted in the presentation of a
       basic EPS calculation in the consolidated financial statements as well as
       a diluted EPS calculation. Basic EPS is computed by dividing net income
       (loss) applicable to common shareholders by the weighted average number
       of common shares outstanding during each period. Diluted EPS is computed
       by dividing net income (loss) applicable to common shareholders by the
       weighted average number of common shares and common share equivalents
       outstanding, if dilutive, during each period. The number of common share
       equivalents outstanding is computed using the treasury stock method.

       Historical earnings per share for the three months ended March 31, 1997
       is based on shares of common stock issued upon consummation of the
       Exchange (Note 1). At March 31, 1997 and 1998, options to purchase
       644,097 and 935,987, respectively, shares of common stock were
       outstanding but were not included in the computation of diluted EPS due
       to the anti-dilutive effect they would have on EPS if converted.




                                        5
<PAGE>   8

                           BRIGHAM EXPLORATION COMPANY

                             NOTES TO THE CONDENSED
                        CONSOLIDATED FINANCIAL STATEMENTS
                                   (UNAUDITED)

6.     REPORTING COMPREHENSIVE INCOME

       In June 1997, the Financial Accounting Standards Board ("FASB") issued
       SFAS No. 130, "Reporting Comprehensive Income." The new standard, which
       is effective for financial statements issued for periods ending after
       December 15, 1997, established standards for reporting, in addition to
       net income, comprehensive income and its components including, as
       applicable, foreign currency items, minimum pension liability adjustments
       and unrealized gains and losses on certain investments in debt and equity
       securities. Upon adoption, the Company is also required to reclassify
       financial statements for earlier periods provided for comparative
       purposes. The Company adopted this standard in the first quarter of 1998.
       There is no difference between the Company's net income as reported and
       comprehensive income.

7.     SEGMENT REPORTING

       In June 1997, the FASB issued SFAS No. 131, "Disclosure about Segments of
       an Enterprise and Related Information", which the Company adopted in the
       first quarter of 1998. The standard established requirements for
       reporting information about operating segments in interim financial
       reports issued to shareholders. It also established standards for related
       disclosures about products and services, geographic areas and major
       customers. Under SFAS No. 131, operating segments are to be determined
       consistent with management's organization and evaluation of financial
       information internally for making operating decisions and assessing
       performance. The disclosure provisions of this standard are not
       applicable for interim periods in the year of adoption. The adoption of
       this new standard is not expected to have a material impact on the
       Company's consolidated balance sheet or statement of operations.




                                        6
<PAGE>   9
                           BRIGHAM EXPLORATION COMPANY

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

Results of Operations

Comparison of three month periods ended March 31, 1997 and March 31, 1998

         Natural gas and oil sales. Natural gas and oil sales increased 47% from
$2.1 million in the first quarter of 1997 to $3.1 million in the first quarter
of 1998. Of this net increase, $2.8 million was attributable to an increase in
production, offset by $1.8 million attributable to a decrease in the average
sales price for natural gas and oil. Production volumes for natural gas
increased 205% from 243 MMcf in the first quarter of 1997 to 740 MMcf in the
first quarter of 1998. The average price received for natural gas decreased 34%
from $3.10 per Mcf in the first quarter of 1997 to $2.05 per Mcf in the first
quarter of 1998. Production volumes for oil increased 82% from 63 MBbls in the
first quarter of 1997 to 115 MBbls in the first quarter of 1998. The average
price received for oil decreased 35% from $21.82 per Bbl in the first quarter of
1997 to $14.15 per Bbl in the first quarter of 1998. Natural gas and oil sales
were increased by production from wells completed during the first quarter of
1998, partially offset by the natural decline of existing production.

         Lease operating expenses. Lease operating expense increased 101% from
$206,000 for the first quarter of 1997 to $414,000 for the first quarter of
1998, while on a per unit of production basis, lease operating expenses for the
same periods decreased 12% from $.33 per Mcfe to $.29 per Mcfe. The increase in
lease operating expenses was primarily due to an increase in the number of
producing wells in the first quarter of 1998 as compared with the same period in
1997. The decrease in the per unit rate was primarily due to an increase in gas
production as a percentage of total equivalent production (39% and 52% for the
first quarters of 1997 and 1998, respectively) since a typical gas well produces
with lower average lease operating costs per unit of production than a typical
oil well.

         Production taxes. Production taxes increased 47% from $127,000 ($.21
per Mcfe) for the first quarter of 1997 to $188,000 ($.13 per Mcfe) for the
first quarter of 1998 as a direct result of increased production volumes. The
effective average production tax rate remained unchanged at 6% of natural gas
and oil sales revenues for each period.

         General and administrative expenses. General and administrative
expenses increased 64% from $702,000 for the first quarter of 1997 to $1.2
million for the first quarter of 1998. This increase was primarily attributable
to the hiring of additional employees to support the Company's increased level
of operational activities. Additionally, office rent, other office expenses and
costs related to the administration of a public corporation increased for the
first quarter of 1998 as compared to the same period for 1997. On a per unit of
production basis, general and administrative expenses decreased 28% from $1.13
per Mcfe for the first quarter of 1997 to $.81 per Mcfe for the first quarter of
1998.

         Depletion of natural gas and oil properties. Depletion of natural gas
and oil properties increased 84% from $687,000 ($1.10 per Mcfe) in the first
quarter of 1997 to $1.3 million ($.89 per Mcfe) in the first quarter of 1998. Of
this net increase, $892,000 was due to the increase in production volumes which
was offset by $312,000 due to a 19% decrease in the depletion rate. The
depletion rate per unit of production decreased due to an increase in natural
gas and oil reserves at lower average capital costs.

         Interest expense. Net interest expense increased 165% from $372,000 in
the first quarter of 1997 to $985,000 in the first quarter of 1998. This
increase was due to a higher average outstanding debt balance in the first
quarter of 1998 partially offset by a decreased effective interest rate. The
weighted average outstanding debt balance increased 179% from $15.3 million in
the first quarter of 1997 to $42.8 million in the first quarter of 1998. The



                                        7
<PAGE>   10

effective annual interest rate decreased 6% from 10% in the first quarter of
1997 to 9.5% in the first quarter of 1998. The increase in the average
outstanding debt balance was primarily a result of increased capital
expenditures related to the Company's exploration activities. At March 31, 1998,
the Company had $50 million in borrowings outstanding under the Credit Facility
which had an annual interest rate of 7.9%. Interest expense increased an
additional $106,000 in the first quarter of 1998 compared to the same period for
1997 due to the amortization of deferred loan fees totaling approximately $1.9
million incurred in connection with the establishment of the Credit Facility in
January 1998. The amortization of these deferred loan fees will continue to be
recognized in the amount of approximately $159,000 per quarter over the term of
the Credit Facility which matures in January 2001.

Liquidity and Capital Resources

         The Company's primary sources of capital have been borrowings
(revolving credit facility and private placement debt), working capital, the
sale of interests in projects and sales of equity. During May 1997, as described
in Note 1 to the Condensed Consolidated Financial Statements included herein,
the Company completed an initial public offering of common stock of the Company
that generated proceeds of approximately $24 million, net of offering costs,
that were used to repay all outstanding debt ($13.3 million) under the Company's
then existing revolving credit facility and to fund capital expenditures.

         In the first quarter of 1998, cash flow used by operations was $2.8
million primarily as a result of the net effects of increased natural gas and
oil revenues, net of production taxes, lease operating expenses and general and
administrative expenses, and an increase in working capital components. Cash
flow used in investing activities was $13.1 million in the first quarter of 1998
primarily as a result of capital expenditures related to exploration activities.
Cash flow provided by financing activities was $16 million, net of deferred loan
fees of $1.9 million, in the first three months of 1998 primarily as a result of
an increase in borrowings under the revolving credit facility to fund the
difference between cash flow from operations and cash flow from investing
activities.

         The Company believes its cash flow from operations and its borrowing
capacity under the Credit Facility will be sufficient to finance its continuing
exploration and production activities during the remainder of 1998.

Forward Looking Information

         The Company may make forward looking statements, oral or written,
including statements in this report, press releases and other filings with the
SEC, relating to the Company's drilling plans, its potential drilling locations,
capital expenditures, use of offering proceeds, the ability of expected sources
of liquidity to support working capital and capital expenditure requirements and
the Company's financial position, business strategy and other plans and
objectives for future operations. Such statements involve risks and
uncertainties, including those relating to the Company's dependence on
exploratory drilling activities, the volatility of natural gas and oil prices,
the risks associated with growth (including the risk of reduced availability of
seismic gathering and drilling services in the face of growing demand), the
substantial capital requirements of the Company's exploration and development
projects, operating hazards and uninsured risks and other factors detailed in
the Company's registration statement and other filings with the SEC. All
subsequent oral and written forward looking statements attributable to the
Company are expressly qualified in their entirety by these factors. The Company
assumes no obligation to update these statements.





                                        8
<PAGE>   11

PART II.      OTHER INFORMATION:

Item 6.       Exhibits and Reports on Form 8-K

              (a)  Exhibits

                      27   Financial Data Schedule


                                   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, in the City of Austin, State of Texas,
on the 14th day of May, 1998.

                                         BRIGHAM EXPLORATION COMPANY


                                    By: /s/ BEN M. BRIGHAM
                                        --------------------------------------
                                        Ben M. Brigham
                                        Chief Executive Officer, President and
                                          Chairman of the Board



                                    By: /s/ CRAIG M. FLEMING
                                        --------------------------------------
                                        Craig M. Fleming
                                        Chief Financial Officer




                                        9
<PAGE>   12

<TABLE>
<CAPTION>
                                                         SEQUENTIALLY
 EXHIBIT                                                   NUMBERED
   NO.               INDEX TO EXHIBITS                       PAGE
   ---               -----------------                       ----
<S>        <C>                                           <C>
   27      Financial Data Schedule                         Tabbed by
                                                          Exhibit No.
</TABLE>


                                       10