<PAGE>   1
 
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
 
                       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 SEPTEMBER 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
 
--------------------------------------------------------------------------------
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 September 30, 1998, the number of shares outstanding of the issuer's
only class of stock was 6,818,999 shares of Common Stock.
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
<PAGE>   2
 
                        PART I -- FINANCIAL INFORMATION
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
ITEM 1. FINANCIAL STATEMENTS
 
                    CONSOLIDATED BALANCE SHEETS -- UNAUDITED
 
                                     ASSETS
 
<TABLE>
<CAPTION>
                                                              SEPTEMBER 30, 1998   MARCH 31, 1998
                                                              ------------------   --------------
<S>                                                           <C>                  <C>
CURRENT ASSETS
  Cash and cash equivalents.................................     $     80,294       $  1,361,693
  Accounts receivable, less allowance for doubtful accounts
     ($7,276 at September 30, 1998 and March 31, 1998,
     respectively)..........................................       10,024,133         13,205,113
  Inventories -- Note B.....................................       25,150,805         24,586,863
  Prepaid expenses and other current assets.................          268,263            193,879
                                                                 ------------       ------------
          Total Current Assets..............................       35,523,495         39,347,548
PROPERTY, PLANT AND EQUIPMENT
  Land......................................................          198,021            198,021
  Buildings and improvements................................        3,144,417          2,882,358
  Machinery and equipment...................................       15,109,396         13,999,439
  Less allowance for depreciation...........................      (10,813,810)       (10,468,859)
                                                                 ------------       ------------
                                                                    7,638,024          6,610,959
OTHER ASSETS
  Cash value of officers' life insurance....................          162,682             80,854
                                                                 ------------       ------------
                                                                 $ 43,324,201       $ 46,039,361
                                                                 ============       ============
 
LIABILITIES AND STOCKHOLDERS' EQUITY
 
CURRENT LIABILITIES
  Trade accounts payable and accrued expenses...............     $  4,319,214       $ 10,925,023
  Current portion of long-term debt.........................          800,000            800,000
  Dividends payable.........................................          477,319            486,886
  Contribution to profit-sharing plan.......................          140,400            280,000
  Income taxes payable......................................           54,257            344,465
  Employee compensation and related expenses................          476,187            600,804
                                                                 ------------       ------------
          Total Current Liabilities.........................        6,267,377         13,437,178
LONG-TERM DEBT, less current portion........................        9,800,000          6,366,666
PROVISION FOR NONPENSION RETIREMENT BENEFITS................          113,000            113,000
DEFERRED INCOME TAXES.......................................          359,560            389,560
STOCKHOLDERS' EQUITY
Common stock:
  Par value $1 per share:
     Authorized 10,000,000 shares; Issued and outstanding
       shares -- 6,818,999 at September 30, 1998 and
       6,491,808 at March 31, 1998..........................        6,818,999          6,491,808
  Additional paid-in capital................................       25,709,766         23,680,628
  Retained earnings.........................................       (5,744,501)        (4,439,479)
                                                                 ------------       ------------
          Total Stockholders' Equity........................       26,784,264         25,732,957
                                                                 ------------       ------------
                                                                 $ 43,324,201       $ 46,039,361
                                                                 ============       ============
</TABLE>
<PAGE>   3
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
                CONSOLIDATED STATEMENTS OF EARNINGS -- UNAUDITED
 
<TABLE>
<CAPTION>
                                              THREE MONTHS ENDED           SIX MONTHS ENDED
                                                 SEPTEMBER 30,               SEPTEMBER 30,
                                           -------------------------   -------------------------
                                              1998          1997          1998          1997
                                           -----------   -----------   -----------   -----------
<S>                                        <C>           <C>           <C>           <C>
Net sales................................  $32,178,289   $36,961,370   $71,101,458   $75,261,802
Costs and expenses
  Costs of goods sold....................   29,311,077    34,065,469    65,312,853    69,131,090
  General, selling and administrative
     costs...............................    1,223,360     1,251,526     2,531,594     2,605,254
  Interest...............................      152,991       112,146       255,713       227,734
                                           -----------   -----------   -----------   -----------
                                            30,687,428    35,429,141    68,100,160    71,964,078
Interest and other income................      (47,145)      (10,091)      (86,326)      (27,270)
                                           -----------   -----------   -----------   -----------
Earnings before federal income taxes.....    1,538,006     1,542,320     3,087,624     3,324,994
Provision (benefit) for federal income
  taxes:
  Current................................      537,922       539,389     1,079,792     1,160,499
  Deferred...............................      (15,000)      (15,000)      (30,000)      (30,000)
                                           -----------   -----------   -----------   -----------
                                               522,922       524,389     1,049,792     1,130,499
                                           -----------   -----------   -----------   -----------
Net earnings.............................  $ 1,015,084   $ 1,017,931   $ 2,037,832   $ 2,194,495
                                           ===========   ===========   ===========   ===========
Average number of common shares
  outstanding:
  Basic..................................    6,818,999     6,800,995     6,818,999     6,800,995
  Diluted................................    6,881,463     6,800,995     6,881,463     6,800,995
Net earnings per share:
  Basic..................................  $      0.15   $      0.15   $      0.30   $      0.32
  Diluted................................  $      0.15   $      0.15   $      0.30   $      0.32
 
Cash dividends declared per common
  share..................................  $      0.07   $     0.075   $     0.145   $     0.145
</TABLE>
<PAGE>   4
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
               CONSOLIDATED STATEMENTS OF CASH FLOWS -- UNAUDITED
 
<TABLE>
<CAPTION>
                                                                   SIX MONTHS ENDED
                                                                    SEPTEMBER 30,
                                                              --------------------------
                                                                 1998           1997
                                                              -----------    -----------
<S>                                                           <C>            <C>
OPERATING ACTIVITIES
  Net earnings..............................................  $ 2,037,832    $ 2,194,495
  Adjustments to reconcile net earnings to cash provided by
     operating activities:
     Depreciation...........................................      344,952        334,202
     Provision for deferred taxes...........................      (30,000)       (30,000)
  Decrease (increase) in operating assets:
     Accounts receivable....................................    3,180,980     (1,098,932)
     Inventories............................................     (563,942)       251,590
     Other..................................................      (74,384)      (313,699)
  Increase (decrease) in operating liabilities:
     Accounts payable and accrued expenses..................   (6,605,809)      (307,269)
     Contribution to profit-sharing plan....................     (139,600)      (117,002)
     Employee compensation and related expenses.............     (124,617)        92,821
     Federal income taxes payable...........................     (290,208)      (256,434)
                                                              -----------    -----------
          NET CASH PROVIDED (USED) BY OPERATING
             ACTIVITIES.....................................   (2,264,796)       749,772
INVESTING ACTIVITIES
  Purchase of property, plant and equipment.................   (1,372,016)    (1,509,095)
  (Increase) decrease in cash value of officers' life
     insurance..............................................      (81,828)       (23,828)
                                                              -----------    -----------
          NET CASH PROVIDED (USED) IN INVESTING
             ACTIVITIES.....................................   (1,453,844)    (1,532,923)
FINANCING ACTIVITIES
  Cash dividends paid.......................................   (1,001,354)      (824,871)
  Principal payments on long-term debt......................     (400,000)      (400,000)
  Proceeds from borrowings of long term debt................    3,833,333      1,900,000
  Exercise of stock options.................................        5,262         13,850
                                                              -----------    -----------
          NET CASH PROVIDED (USED) IN FINANCING
             ACTIVITIES.....................................    2,437,241        688,979
                                                              -----------    -----------
INCREASE (DECREASE) IN CASH AND CASH EQUIVALENTS............   (1,281,399)       (94,172)
  Cash and cash equivalents at beginning of period..........    1,361,693        168,245
                                                              -----------    -----------
CASH AND CASH EQUIVALENTS AT END OF PERIOD..................  $    80,294    $    74,073
                                                              ===========    ===========
</TABLE>
<PAGE>   5
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
                     NOTES TO QUARTERLY REPORT -- UNAUDITED
                     THREE MONTHS ENDED SEPTEMBER 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.
<PAGE>   6
 ITEM 2. MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND RESULTS
        OF OPERATIONS
 
SIX MONTHS ENDED SEPTEMBER 30, 1998 COMPARED TO SIX MONTHS ENDED SEPTEMBER 30,
1997
 
     During the six month period ended September 30, 1998, sales, costs of goods
sold and gross profit were approximately even with the respective amounts
recorded during the six month period ended September 30, 1997. Decreases in
sales and gross profit produced by the Company's tubular division were offset by
increases in sales and gross profit generated by coil operations. Operations of
the tubular division were adversely affected by decreased market demand for
tubular products which reduced volume and margins. The Company's coil operations
benefited from increased volume and margins.
 
     Interest expense during the 1998 period increased $27,979 from the amount
recorded during the 1997 period. This increase was primarily related to interest
paid on additional borrowings pursuant to the Company's line of credit
arrangement with a bank.
 
     Interest and other income increased $59,056 from the amount recorded in the
1997 period. This increase was primarily related to other income associated with
an increase in the cash surrender value of officers' life insurance.
 
 THREE MONTHS ENDED SEPTEMBER 30, 1998 COMPARED TO THREE MONTHS ENDED SEPTEMBER
                                    30, 1997
 
     During the three months ended September 30, 1998, sales, costs of goods
sold and gross profit declined $4,783,081, $4,754,392 and $28,689, respectively,
from comparable amounts recorded during the three months ended September 30,
1997. The decreases in sales and costs of goods sold were primarily related to
the Company's tubular operations. Decreased market demand for tubular products
reduced volume and margins during the 1998 quarter. A decline in gross profit
earned by tubular operations was partially offset by an increase in gross profit
earned by the Company's coil operations. Steady demand for coil products and
reduced cost of material benefited coil operations which reflected improved
margins during the 1998 quarter.
 
     Interest expense increased $40,845 from the amount recorded during the 1997
quarter. This increase was primarily related to interest paid on additional
borrowings pursuant to the Company's line of credit arrangement with a bank.
 
     Interest and other income increased $37,054 from the amount recorded during
the 1997 quarter. This increase was primarily related to other income associated
with an increase in the cash surrender value of officers' life insurance.
 
FINANCIAL POSITION, LIQUIDITY AND CAPITAL RESOURCES
 
     The Company remained in a strong, liquid position at September 30, 1998.
Current ratios were 5.7 and 2.9 at September 30, 1998 and March 31, 1998,
respectively. Working capital was $29,256,118 at September 30, 1998 and
$25,910,370 at March 31, 1998.
 
     The Company has a credit arrangement with a bank that 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 September 30, 1998, the Company had
borrowings outstanding under the revolving facility of $7 million. The term
facility includes borrowings of $1.2 million from a 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 September 30, 1998, the principal amount of
indebtedness outstanding under the term facility was $3.6 million.
<PAGE>   7
 
EFFECT OF YEAR 2000 ISSUE
 
     The Year 2000 issue is the result of computer programming being written
using two digits rather than four to define the applicable year. Any of the
Company's systems, as well as those of key vendors, payors and customers, that
have date sensitive logic may interpret a date using "00" as the year 1900
rather than 2000. This may cause inaccurate processing or possible system
failure and may potentially disrupt operations. This disruption may result in,
among other things, a temporary inability to process transactions, send bills
for services or engage in similar normal business activities.
 
     In 1998, the Company completed an assessment of the readiness of its
internal computer systems and related applications to accommodate date-sensitive
information relating to the year 2000 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.
 
     The Company will continue to analyze systems and services that utilize
date-embedded codes that may experience operational problems when the year 2000
is reached. The Company will continue to communicate with its suppliers,
third-party payors and customers to coordinate Year 2000 compliance. Because the
ability of these third parties to address their Year 2000 issues is outside the
Company's control, the failure of third parties to adequately address their
respective Year 2000 issues may have a material adverse effect on the Company's
results of operations and financial condition.
 
     The foregoing statements are intended to be and are hereby designated "Year
2000 Readiness Disclosure" statements within the meaning of the Year 2000
Information and Readiness Disclosure Act.
 
ITEM 3. QUANTITATIVE AND QUALITATIVE DISCLOSURES ABOUT MARKET RISK
 
     Not applicable.
<PAGE>   8
 
                          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
 
     At the Annual Meeting of Shareholders held on August 21, 1998, the
Company's shareholders elected eight directors to the Company's Board of
Directors. The number of shares voted for and withheld with respect to the
election of each director was as follows:
 
<TABLE>
<CAPTION>
                         NAME                          SHARES VOTED FOR   SHARES WITHHELD
                         ----                          ----------------   ---------------
<S>                                                    <C>                <C>
Jack Friedman.........................................    6,417,174            9,915
Harold Friedman.......................................    6,417,174            9,915
William E. Crow.......................................    6,417,174            9,915
Charles W. Hall.......................................    6,417,174            9,915
Alan M. Rauch.........................................    6,417,174            9,915
Hershel M. Rich.......................................    6,417,174            9,915
Henry Spira...........................................    6,417,174            9,915
Kirk K. Weaver........................................    6,417,174            9,915
</TABLE>
 
ITEM 5. OTHER INFORMATION
 
     Not applicable
 
ITEM 6. EXHIBITS AND REPORTS ON FORM 8-K
 
     a). Exhibits
 
        27.1 -- Financial Data Schedule
 
     b). Reports on Form 8-K
 
          None
<PAGE>   9
 
                                   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 November 16, 1998                      By        /s/  BEN HARPER
 
                                            ------------------------------------
                                                  Ben Harper, Senior Vice
                                                     President-Finance
                                                 (Chief Accounting Officer)
 
Date November 16, 1998                      By      /s/  HAROLD FRIEDMAN
 
                                            ------------------------------------
                                                Harold Friedman, Vice Chairman
                                            of the Board
<PAGE>   10
 
                               INDEX TO EXHIBITS
 
<TABLE>
<CAPTION>
        EXHIBIT
         NUMBER                                  DESCRIPTION
        -------                                  -----------
<C>                      <S>
         27.1            -- Financial Data Schedule.
</TABLE>