<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 DECEMBER 31, 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 December 31, 1998, the number of shares outstanding of the issuer's only
class of stock was 6,826,387 shares of Common Stock.
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<PAGE>   2
 
                        PART I -- FINANCIAL INFORMATION
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
ITEM 1. FINANCIAL STATEMENTS
 
                    CONSOLIDATED BALANCE SHEETS -- UNAUDITED
 
                                     ASSETS
 
<TABLE>
<CAPTION>
                                                              DECEMBER 31, 1998   MARCH 31, 1998
                                                              -----------------   --------------
<S>                                                           <C>                 <C>
CURRENT ASSETS
  Cash and cash equivalents.................................    $     57,336       $  1,361,693
  Accounts receivable, less allowance for doubtful accounts
     ($7,276 at December 31, 1998 and March 31, 1998,
     respectively)..........................................       8,237,292         13,205,113
  Inventories -- Note B.....................................      23,570,040         24,586,863
  Prepaid expenses and other current assets.................         590,919            193,879
                                                                ------------       ------------
          Total Current Assets..............................      32,455,587         39,347,548
PROPERTY, PLANT AND EQUIPMENT
  Land......................................................         198,021            198,021
  Buildings and improvements................................       3,233,091          2,882,358
  Machinery and equipment...................................      15,406,919         13,999,439
  Less allowance for depreciation...........................     (10,986,287)       (10,468,859)
                                                                ------------       ------------
                                                                   7,851,744          6,610,959
OTHER ASSETS
  Cash value of officers' life insurance....................         193,666             80,854
                                                                ------------       ------------
                                                                $ 40,500,997       $ 46,039,361
                                                                ============       ============
 
LIABILITIES AND STOCKHOLDERS' EQUITY
 
CURRENT LIABILITIES
  Trade accounts payable and accrued expenses...............    $  4,729,756       $ 10,925,023
  Current portion of long-term debt.........................         800,000            800,000
  Dividends payable.........................................         477,837            486,886
  Contribution to profit-sharing plan.......................         210,600            280,000
  Income taxes payable......................................              --            344,465
  Employee compensation and related expenses................         223,967            600,804
                                                                ------------       ------------
          Total Current Liabilities.........................       6,442,160         13,437,178
LONG-TERM DEBT, less current portion........................       6,600,000          6,366,666
PROVISION FOR NONPENSION RETIREMENT BENEFITS................         113,000            113,000
DEFERRED INCOME TAXES.......................................         344,560            389,560
STOCKHOLDERS' EQUITY
Common stock:
  Par value $1 per share:
     Authorized 10,000,000 shares; Issued and outstanding
       shares -- 6,826,387 at December 31, 1998 and
       6,491,808 at
       March 31, 1998.......................................       6,826,387          6,491,808
  Additional paid-in capital................................      25,715,528         23,680,628
  Retained earnings.........................................      (5,540,638)        (4,439,479)
                                                                ------------       ------------
          Total Stockholders' Equity........................      27,001,277         25,732,957
                                                                ------------       ------------
                                                                $ 40,500,997       $ 46,039,361
                                                                ============       ============
</TABLE>
 
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<PAGE>   3
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
                CONSOLIDATED STATEMENTS OF EARNINGS -- UNAUDITED
 
<TABLE>
<CAPTION>
                                             THREE MONTHS ENDED           NINE MONTHS ENDED
                                                DECEMBER 31,                 DECEMBER 31,
                                          -------------------------   --------------------------
                                             1998          1997          1998           1997
                                          -----------   -----------   -----------   ------------
<S>                                       <C>           <C>           <C>           <C>
Net sales...............................  $26,938,495   $34,300,676   $98,039,953   $109,562,478
Costs and expenses:
  Costs of goods sold...................   24,806,991    31,533,234    90,119,844    100,664,324
  General, selling and administrative
     costs..............................    1,035,859     1,151,863     3,567,453      3,757,117
  Interest..............................      112,042       116,799       367,755        344,533
                                          -----------   -----------   -----------   ------------
                                           25,954,892    32,801,896    94,055,052    104,765,974
Interest and other income...............      (49,277)      (30,233)     (135,603)       (57,503)
                                          -----------   -----------   -----------   ------------
Earnings before federal income taxes....    1,032,880     1,529,013     4,120,504      4,854,007
Provision (benefit) for federal income
  taxes:
  Current...............................      366,180       534,865     1,445,972      1,695,364
  Deferred..............................      (15,000)      (15,000)      (45,000)       (45,000)
                                          -----------   -----------   -----------   ------------
                                              351,180       519,865     1,400,972      1,650,364
                                          -----------   -----------   -----------   ------------
Net earnings............................  $   681,700   $ 1,009,148   $ 2,719,532   $  3,203,643
                                          ===========   ===========   ===========   ============
Average number of common shares
  outstanding:
  Basic.................................    6,826,387     6,807,527     6,826,387      6,799,920
  Diluted...............................    6,863,948     6,956,754     6,863,948      6,940,162
Net earnings per share:
  Basic.................................  $      0.10   $      0.15   $      0.40   $       0.47
  Diluted...............................  $      0.10   $      0.15   $      0.40   $       0.46
 
Cash dividends declared per common
  share.................................  $      0.07   $     0.075   $     0.215   $       0.22
</TABLE>
 
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<PAGE>   4
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
               CONSOLIDATED STATEMENTS OF CASH FLOWS -- UNAUDITED
 
<TABLE>
<CAPTION>
                                                                  NINE MONTHS ENDED
                                                                     DECEMBER 31,
                                                              --------------------------
                                                                 1998           1997
                                                              -----------    -----------
<S>                                                           <C>            <C>
OPERATING ACTIVITIES
  Net earnings..............................................  $ 2,719,532    $ 3,203,643
  Adjustments to reconcile net earnings to cash provided by
     operating activities:
     Depreciation...........................................      517,428        506,678
     Provision for deferred taxes...........................      (45,000)       (45,000)
  Decrease (increase) in operating assets:
     Accounts receivable....................................    4,967,821       (122,286)
     Inventories............................................    1,016,823       (351,901)
     Other..................................................     (397,040)      (214,025)
  Increase (decrease) in operating liabilities:
     Accounts payable and accrued expenses..................   (6,195,267)     2,387,514
     Contribution to profit-sharing plan....................      (69,400)       (54,503)
     Employee compensation and related expenses.............     (376,837)       (19,239)
     Federal income taxes...................................     (344,465)      (256,434)
                                                              -----------    -----------
          NET CASH PROVIDED (USED) BY OPERATING
             ACTIVITIES.....................................    1,793,595      5,034,447
INVESTING ACTIVITIES
  Purchase of property, plant and equipment.................   (1,758,212)    (1,858,205)
  Decrease (increase) in cash value of officers' life
     insurance..............................................     (112,812)       (24,177)
                                                              -----------    -----------
          NET CASH PROVIDED (USED) IN INVESTING
             ACTIVITIES.....................................   (1,871,024)    (1,882,382)
FINANCING ACTIVITIES
  Cash dividends paid.......................................   (1,478,674)    (1,310,801)
  Principal payments on long-term debt......................   (3,600,000)    (1,600,000)
  Proceeds from borrowings of long term debt................    3,833,333      2,733,333
  Exercise of stock options.................................       18,413         50,767
                                                              -----------    -----------
          NET CASH PROVIDED (USED) IN FINANCING
             ACTIVITIES.....................................   (1,226,928)      (126,701)
INCREASE (DECREASE) IN CASH AND CASH EQUIVALENTS............   (1,304,357)     3,025,364
  Cash and cash equivalents at beginning of period..........    1,361,693        168,245
                                                              -----------    -----------
CASH AND CASH EQUIVALENTS AT END OF PERIOD..................  $    57,336    $ 3,193,609
                                                              ===========    ===========
</TABLE>
 
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<PAGE>   5
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
                     NOTES TO QUARTERLY REPORT -- UNAUDITED
                      THREE MONTHS ENDED DECEMBER 31, 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.
 
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<PAGE>   6
 
ITEM 2. MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND RESULTS
        OF OPERATIONS
 
 NINE MONTHS ENDED DECEMBER 31, 1998 COMPARED TO NINE MONTHS ENDED DECEMBER 31,
                                      1997
 
     During the nine month period ended December 31, 1998, sales, costs of goods
sold and gross profit declined $11,522,525, $10,544,480 and $978,045,
respectively, from the comparable amounts recorded during the nine month period
ended December 31, 1997. During the 1998 period, increases in sales, costs of
goods sold and gross profit attributable to coil operations were more than
offset by substantial declines in the comparable financial categories associated
with tubular operations. Tubular operations were adversely affected by a
significant downturn of the energy sector of the U.S. economy which had the
effect of reducing demand for tubular products and generating intense
competition for available sales.
 
     Interest and other income during the 1998 period increased $78,100 from the
amount recorded during the 1997 period. This increase was primarily associated
with other income related to an increase in the cash surrender value of
officers' life insurance.
 
     Federal income tax declined $249,392 from the comparable amount recorded in
the 1997 period. This decline was related to the decrease in income before taxes
as the effective tax rates were the same for both periods.
 
THREE MONTHS ENDED DECEMBER 31, 1998 COMPARED TO THREE MONTHS ENDED DECEMBER 31,
                                      1997
 
     During the quarter ended December 31, 1998, sales, costs of goods sold and
gross profit declined $7,362,181, $6,726,243 and $635,938, respectively, from
comparable amounts recorded during the quarter ended December 31, 1997. The
declines in sales and costs of goods sold were primarily related to a decline in
tubular operations. An increase in gross profit attributable to coil operations
was more than offset by a substantial decrease in gross profit associated with
tubular operations. Tubular operations were adversely affected by a significant
downturn in the energy sector of the U.S. economy which had the effect of
reducing demand for tubular products and generating intense competition for
available sales.
 
     General, selling and administrative costs declined $116,004 from the
comparable amount recorded during the 1997 quarter. This decline was primarily
related to a decline in incentive bonuses based on net earnings.
 
     Interest and other income increased $19,044 from the amount recorded during
the 1997 quarter. This increase was primarily associated with other income
related to an increase in the cash value of officers' life insurance.
 
     Federal income taxes declined $168,685 from the comparable amount recorded
in the 1997 quarter. This decline was related to the decrease in income before
taxes during the 1998 quarter as the effective tax rates were the same for both
quarters.
 
FINANCIAL POSITION, LIQUIDITY AND CAPITAL RESOURCES
 
     The Company remained in a strong, liquid position at December 31, 1998.
Current ratios were 5.0 and 2.9 at December 31, 1998 and March 31, 1998,
respectively. Working capital was $26,013,427 at December 31, 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 December 31, 1998, the Company had
borrowings outstanding under the revolving facility of $4 million. The term
facility includes borrowings of $1.2 million from a previous term note and prior
to December 31, 1998, also provided for additional advances up to $3.5 million,
all of which converted 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
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<PAGE>   7
 
remain in effect for the entire term of the term facility. As of December 31,
1998, the principal amount of indebtedness outstanding under the term facility
was $3.4 million.
 
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 suppliers, third party payors and
customers, having 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.
 
                                        6
<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
 
     None
 
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
 
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<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 February 12, 1999                      By        /s/  BEN HARPER
 
                                            ------------------------------------
                                                  Ben Harper, Senior Vice
                                                     President-Finance
                                                 (Chief Accounting Officer)
 
Date February 12, 1999                      By      /s/  HAROLD FRIEDMAN
 
                                            ------------------------------------
                                                Harold Friedman, Vice Chairman
 
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<PAGE>   10
 
                               INDEX TO EXHIBITS
 
<TABLE>
<CAPTION>
        EXHIBIT
         NUMBER                                  DESCRIPTION
        -------                                  -----------
<C>                      <S>
         27.1            -- Financial Data Schedule.
</TABLE>