<PAGE>   1
 
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                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 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 JUNE 30, 1998
 
                                       OR
 
[ ] TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(D) OF THE SECURITIES EXCHANGE
    ACT OF 1934 FROM THE TRANSITION PERIOD FROM                   TO
 
                         COMMISSION FILE NUMBER 1-7521
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
             (Exact name of registrant as specified in its charter)
 
<TABLE>
<S>                                            <C>
                    TEXAS                                             74-1504405
       (State or other jurisdiction of                     (I.R.S. Employer Identification
        incorporation or organization)                                 Number)
</TABLE>
 
                 4001 HOMESTEAD ROAD, HOUSTON, TEXAS 77028-5585
                (Address of principal executive office zip code)
       Registrant's telephone number, including area code (713) 672-9433
 
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Former name, former address and former fiscal year, of changed since last report
 
     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, and (2) has been subject to such filing
requirements for the past 90 days.
 
                               Yes  X     No
 
     At June 30, 1998, the number of shares outstanding of the issuer's only
class of stock was 6,818,999 shares of Common Stock.
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<PAGE>   2
 
                        PART I -- FINANCIAL INFORMATION
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
                    CONSOLIDATED BALANCE SHEETS -- UNAUDITED
 
                                     ASSETS
 
<TABLE>
<CAPTION>
                                                              JUNE 30, 1998    MARCH 31, 1998
                                                              -------------    --------------
<S>                                                           <C>              <C>
CURRENT ASSETS
  Cash and cash equivalents.................................  $      96,360    $    1,361,693
  Accounts receivable, less allowance for doubtful accounts
     ($7,276 at June 30, 1998 and March 31, 1998,
     respectively)..........................................     14,001,004        13,205,113
  Inventories -- Note B.....................................     24,130,125        24,586,863
  Prepaid expenses and other current assets.................        247,196           193,879
                                                              -------------    --------------
          Total Current Assets..............................     38,474,685        39,347,548
PROPERTY, PLANT AND EQUIPMENT
  Land......................................................        198,021           198,021
  Buildings and improvements................................      3,077,678         2,882,358
  Machinery and equipment...................................     14,752,758        13,999,439
  Less allowance for depreciation...........................    (10,641,335)      (10,468,859)
                                                              -------------    --------------
                                                                  7,387,122         6,610,959
OTHER ASSETS
  Cash value of officers' life insurance....................        119,417            80,854
                                                              -------------    --------------
                                                              $  45,981,224    $   46,039,361
                                                              =============    ==============
 
LIABILITIES AND STOCKHOLDERS' EQUITY
 
CURRENT LIABILITIES
  Trade accounts payable and accrued expenses...............  $   9,791,626    $   10,925,023
  Current portion of long-term debt.........................        800,000           800,000
  Dividends payable.........................................        511,419           486,886
  Contribution to profit-sharing plan.......................         70,200           280,000
  Income taxes payable......................................        576,335           344,465
  Employee compensation and related expenses................        497,311           600,804
                                                              -------------    --------------
          Total Current Liabilities.........................     12,246,891        13,437,178
LONG-TERM DEBT, less current portion........................      7,000,000         6,366,666
PROVISION FOR NONPENSION RETIREMENT BENEFITS................        113,000           113,000
DEFERRED INCOME TAXES.......................................        374,560           389,560
STOCKHOLDERS' EQUITY
Common stock:
  Par value $1 per share:
     Authorized 10,000,000 shares; Issued and outstanding
       shares -- 6,818,999 at June 30, 1998 and 6,491,808 at
       March 31, 1998.......................................      6,818,999         6,491,808
  Additional paid-in capital................................     25,710,432        23,680,628
  Retained earnings.........................................     (6,282,658)       (4,439,479)
                                                              -------------    --------------
          Total Stockholders' Equity........................     26,246,773        25,732,957
                                                              -------------    --------------
                                                              $  45,981,224    $   46,039,361
                                                              =============    ==============
</TABLE>
 
                                        1
<PAGE>   3
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
                CONSOLIDATED STATEMENTS OF EARNINGS -- UNAUDITED
 
<TABLE>
<CAPTION>
                                                              THREE MONTHS ENDED JUNE 30,
                                                              ----------------------------
                                                                  1998            1997
                                                              ------------    ------------
<S>                                                           <C>             <C>
Net sales...................................................  $38,923,169     $38,300,432
Costs and expenses
  Costs of goods sold.......................................   36,001,776      35,065,621
  General, selling and administrative costs.................    1,308,234       1,353,728
  Interest..................................................      102,722         115,588
                                                              -----------     -----------
                                                               37,412,732      36,534,937
Interest and other income...................................      (39,181)        (17,179)
                                                              -----------     -----------
Earnings before federal income taxes........................    1,549,618       1,782,674
Provision (benefit) for federal income taxes:
  Current...................................................      541,870         621,110
  Deferred..................................................      (15,000)        (15,000)
                                                              -----------     -----------
                                                                  526,870         606,110
                                                              -----------     -----------
Net earnings................................................  $ 1,022,748     $ 1,176,564
                                                              ===========     ===========
Average number of common shares outstanding:
  Basic.....................................................    6,818,999       6,793,218
  Diluted...................................................    6,958,736       6,793,218
Net earnings per share:
  Basic.....................................................  $      0.15     $      0.17
  Diluted...................................................  $      0.15     $      0.17
 
Cash dividends declared per common share....................  $     0.075     $      0.07
</TABLE>
 
                                        2
<PAGE>   4
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
               CONSOLIDATED STATEMENTS OF CASH FLOWS -- UNAUDITED
 
<TABLE>
<CAPTION>
                                                              THREE MONTHS ENDED JUNE 30,
                                                              ----------------------------
                                                                  1998            1997
                                                              ------------    ------------
<S>                                                           <C>             <C>
OPERATING ACTIVITIES
  Net earnings..............................................  $ 1,022,748     $ 1,176,564
  Adjustments to reconcile net income to cash provided by
     operating activities:
     Depreciation...........................................      172,476         166,026
     Provision for deferred taxes...........................      (15,000)        (15,000)
  Decrease (increase) in operating assets:
     Accounts receivable....................................     (795,891)     (1,727,438)
     Inventories............................................      456,738      (2,256,649)
     Other current assets...................................      (53,317)        (66,810)
  Increase (decrease) in operating liabilities:
     Accounts payable and accrued expenses..................   (1,133,397)      3,624,540
     Contribution to profit-sharing plan payable............     (209,800)       (179,501)
     Employee compensation and related expenses.............     (103,493)        170,228
     Federal income taxes payable...........................      231,870         376,264
                                                              -----------     -----------
NET CASH PROVIDED (USED) BY OPERATING ACTIVITIES............     (427,066)      1,268,224
INVESTING ACTIVITIES
  Purchase of property, plant and equipment.................     (948,639)       (313,069)
  Increase in cash surrender value of officers' life
     insurance..............................................      (38,563)         (6,919)
                                                              -----------     -----------
          NET CASH USED IN INVESTING ACTIVITIES.............     (987,202)       (319,988)
FINANCING ACTIVITIES
  Cash dividends paid.......................................     (489,660)       (371,575)
  Principal payments on long-term debt......................     (200,000)       (200,000)
  Proceeds from borrowings of long-term debt................      833,333              --
  Cash received from exercised stock options................        5,262              --
                                                              -----------     -----------
          NET CASH PROVIDED (USED) IN FINANCING
             ACTIVITIES.....................................      148,935        (571,575)
                                                              -----------     -----------
INCREASE (DECREASE) IN CASH AND CASH EQUIVALENTS............   (1,265,333)        376,661
  Cash and cash equivalents at beginning of period..........    1,361,693         168,245
                                                              -----------     -----------
CASH AND CASH EQUIVALENTS AT END OF PERIOD..................  $    96,360     $   544,906
                                                              ===========     ===========
</TABLE>
 
                                        3
<PAGE>   5
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
                     NOTES TO QUARTERLY REPORT -- UNAUDITED
                        THREE MONTHS ENDED JUNE 30, 1998
 
NOTE A -- BASIS OF PRESENTATION
 
     The accompanying unaudited condensed, consolidated financial statements
have been prepared in accordance with the instructions to Form 10-Q and 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 accruals) considered
necessary for a fair presentation have been included. For further information,
refer to the financial statements and footnotes included in the Company's annual
report on Form 10-K for the year ended March 31, 1998.
 
NOTE B -- INVENTORIES
 
     Coil inventory consists primarily of raw materials. Tubular inventory is
comprised of both raw materials and finished goods.
 
NOTE C -- EARNINGS PER SHARE
 
     In February 1997, the Financial Accounting Standards Board issued Statement
of Financial Accounting Standards No. 128, Earnings Per Share, which is required
to be adopted for financial statements issued for periods ending after December
31, 1997. This new standard did not have a significant effect on earnings per
share. The difference between the average number of shares outstanding used for
basic and diluted earnings per share is attributable to stock options.
Applicable per share amounts have been adjusted to give effect to stock
dividends.
 
                                        4
<PAGE>   6
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
                    MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                 FINANCIAL CONDITION AND RESULTS OF OPERATIONS
 
 THREE MONTHS ENDED JUNE 30, 1998 COMPARED TO THREE MONTHS ENDED JUNE 30, 1997
 
     During the three months ended June 30, 1998, net sales and costs of goods
sold increased $622,737 and $936,155, respectively, from the comparable amounts
recorded during the three months ended June 30, 1997, the effect of which
reduced gross profit by $313,418. Improved gross profit generated by the coil
operations was more than offset by a decline in gross profit earned by tubular
operations. The Company's coil operations benefited from increased sales
associated primarily with volume and improved margins. During the 1998 period,
the Company continued its effort to aggressively market these products to
increase its market share. Market demand for tubular products declined in the
1998 period, the effect of which adversely affected tubular operations and
reduced volume and margins.
 
     Interest expense declined $12,866 in the 1998 period compared to the 1997
period. This decline was primarily caused by reductions in term debt associated
with operations. Interest on term debt related to construction costs was
capitalized.
 
     Interest and other income in the 1998 period increased $22,002 from the
amount recorded in the 1997 period. This increase was primarily related to
income associated with an increase in the cash surrender value of officers' life
insurance.
 
     Federal income taxes declined $79,240 as a result of the decline in
earnings before taxes. The effective tax rates were the same for both periods.
 
FINANCIAL POSITION, LIQUIDITY AND CAPITAL RESOURCES
 
     The Company remained in a strong, liquid position at June 30, 1998. Current
ratios were 3.1 and 2.9 at June 30, 1998 and March 31, 1998, respectively.
Working capital was $26,227,794 at June 30, 1998 and $25,910,370 at March 31,
1998.
 
     The Company has a credit arrangement with a bank which provides for a
revolving line of credit facility (the "revolving facility") and a term credit
facility (the "term facility"). Pursuant to the revolving facility which expires
April 1, 2000, the Company may borrow up to $8 million at an interest rate no
greater than the bank's prime rate. At June 30, 1998, the Company had borrowings
outstanding under the revolving facility of $4 million. The term facility
includes borrowings of $1.2 million from the previous term note and also
provides for additional advances up to $3.5 million, all of which convert to a
term loan on December 31, 1998. The amount outstanding under the term facility
bears interest at a stated rate of LIBOR plus 1.25% and requires quarterly
principal payments of $200,000 plus accrued interest through March 1, 2003. In
July 1997, the Company entered into a swap transaction with the bank pursuant to
which it exchanged the term facility's LIBOR-based interest rate obligation for
a fixed interest rate obligation of 8% to remain in effect for the entire term
of the term facility. As of June 30, 1998, the principal amount of indebtedness
outstanding under the term facility was $3.8 million.
 
YEAR 2000 ISSUE
 
     With the turn of the century, time sensitive software using two digits may
not identify the year 2000, which could result in system failure and
miscalculations disrupting the ability to conduct normal business operations.
The Company completed an assessment of its information systems for year 2000
compliance during 1998 and developed a plan to resolve all major issues by the
end of 1999. As a result, the year 2000 issue is not expected to pose
significant operational or financial problems for the Company.
 
                                        5
<PAGE>   7
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
                          QUARTER ENDED JUNE 30, 1998
 
PART II -- OTHER INFORMATION
 
ITEM 1. LEGAL PROCEEDINGS
 
     Not applicable
 
ITEM 2. CHANGES IN SECURITIES
 
     a). Not applicable
 
     b). Not applicable
 
     c). Not applicable
 
     d). Not applicable
 
ITEM 3. DEFAULTS UPON SENIOR SECURITIES
 
     a). Not applicable
 
     b). Not applicable
 
ITEM 4. SUBMISSION OF MATTERS TO A VOTE OF SECURITY HOLDERS
 
     None
 
ITEM 5. OTHER INFORMATION
 
     Not applicable
 
ITEM 6. EXHIBITS AND REPORTS ON FORM 8-K
 
     a). Exhibits
 
<TABLE>
<C>                      <S>
         27.1            -- Financial Data Schedule
</TABLE>
 
     b). Reports on Form 8-K
 
          None
 
                                        6
<PAGE>   8
 
                                   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.
 
                                            FRIEDMAN INDUSTRIES, INCORPORATED
 
Date August 14, 1998                        By        /s/  BEN HARPER
 
                                            ------------------------------------
                                                  Ben Harper, Senior Vice
                                                     President-Finance
                                                 (Chief Accounting Officer)
 
Date August 14, 1998                        By      /s/  HAROLD FRIEDMAN
 
                                            ------------------------------------
                                              Harold Friedman, Vice Chairman of
                                                         the Board
 
                                        7
<PAGE>   9
                     INDEPENDENT ACCOUNTANTS' REVIEW REPORT

Board of Directors
Friedman Industries, Incorporated

We have reviewed the accompanying consolidated balance sheet of Friedman
Industries, Incorporated, as of June 30, 1998, and the related consolidated
statements of earnings and cash flows for the three-month periods ended June
30, 1998 and 1997. These financial statements are the responsibility of the
Company's management.

We conducted our reviews in accordance with standards established by the
American Institute of Certified Public Accountants. A review of interim
financial information consists principally of applying analytical procedures to
financial data and making inquiries of persons responsible for financial and
accounting matters. It is substantially less in scope than an audit conducted
in accordance with generally accepted auditing standards, which will be
performed for the full year with the objective of expressing an opinion
regarding the financial statements taken as a whole. Accordingly, we do not
express such an opinion.

Based on our reviews, we are not aware of any material modifications that
should be made to the accompanying condensed consolidated financial statements
referred to above for them to be in conformity with generally accepted
accounting principles.

We have previously audited, in accordance with generally accepted auditing
standards, the consolidated balance sheet of Friedman Industries, Incorporated,
as of March 31, 1998, and the related consolidated statements of earnings,
stockholders' equity and cash flows for the year then ended (not presented
herein) and in our report dated May 26, 1998, we expressed an unqualified
opinion on those consolidated financial statements. In our opinion, the
information set forth in the accompanying condensed consolidated balance sheet
as of March 31, 1998, is fairly stated, in all material respects, in relation
to the consolidated balance sheet from which it has been derived.



                                        /s/ Ernst & Young LLP


Houston, Texas
August 11, 1998
<PAGE>   10
 
                               INDEX TO EXHIBITS
 
<TABLE>
<CAPTION>
        EXHIBIT
         NUMBER                                  DESCRIPTION
        -------                                  -----------
<C>                      <S>
         27.1            -- Financial Data Schedule.

</TABLE>