<PAGE>
 
                UNITED STATES SECURITIES AND EXCHANGE COMMISSION
                             Washington, DC  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 July 31, 1998

 _____
|_____|  Transition Report Pursuant to Section 13 or 15(d) of the Securities
         Exchange Act of 1934

For the transition period from ____________ to _____________

Commission file number 0-5286


                        KEWAUNEE SCIENTIFIC CORPORATION
                        -------------------------------

             (Exact name of registrant as specified in its charter)

           Delaware                                              38-0715562
--------------------------------------------------------------------------------

(State or other jurisdiction of                                (I.R.S.Employer
 incorporation or organization)                              Identification No.)

2700 West Front Street
Statesville, North Carolina                                         28677
--------------------------------------------------------------------------------
(Address of principal executive offices)                         (Zip Code)

                                (704) 873-7202
                         ----------------------------

              (Registrant's telephone number, including area code)

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 (or for such shorter period that the registrant was
required to file such reports), and (2) has been subject to such filing
requirements for the past 90 days.  Yes    X     No 
                                         -----      -----

As of August 28, 1998, the Registrant had outstanding 2,428,796 shares of Common
Stock.

Pages: This report, excluding exhibits, contains 14 pages numbered sequentially
from this cover page.
<PAGE>
 
                        KEWAUNEE SCIENTIFIC CORPORATION

                               INDEX TO FORM 10-Q

                  FOR THE QUARTERLY PERIOD ENDED JULY 31, 1998

<TABLE>
<CAPTION>
                                                                     Page Number
                                                                     -----------
<S>                                                                  <C>

PART I.  FINANCIAL INFORMATION
------------------------------
 
Item 1. Financial Statements
 
        Condensed Statements of Operations -
         Three months ended July 31, 1998 and 1997                         3
 
        Condensed Balance Sheets - July 31, 1998
         and April 30, 1998                                                4
 
        Condensed Statements of Cash Flows -
          Three months ended July 31, 1998 and 1997                        5
 
        Notes to Condensed Financial Statements                            6
 
 
Item 2. Management's Discussion and Analysis of Financial
         Condition and Results of Operations                               7
 
Review by Independent Accountants                                         11
 
Report by Independent Accountants                                         12
 

PART II.  OTHER INFORMATION
---------------------------

Item 4. Submission of Matters to a Vote of
           Security Holders                                               13

Item 6. Exhibits and Reports on Form 8-K                                  13


SIGNATURE                                                                 14
---------                                                      
</TABLE> 

                                       2
<PAGE>
 
                         Part 1. Financial Information

Item 1. Financial Statements


                        Kewaunee Scientific Corporation
                       Condensed Statements of Operations
                                  (Unaudited)
                     (in thousands, except per share data)

<TABLE>
<CAPTION>
                                            Three months ended
                                                 July 31
                                      ------------------------------
                                       1998                   1997
                                      -------                -------
<S>                                   <C>                    <C>
Net sales                             $19,624                $17,662

Costs of products sold                 15,202                 13,524
                                      -------                -------

Gross profit                            4,422                  4,138

Operating expenses                      3,123                  3,166
                                      -------                -------

Operating earnings                      1,299                    972

Interest expense                          (12)                   (52)

Other income (expense), net               (10)                    11
                                      -------                -------

Earnings before income taxes            1,277                    931

Income tax expense                        511                    372
                                      -------                -------

Net earnings                             $766                   $559
                                      =======                =======

Net earnings per share
      Basic                             $0.32                  $0.24
      Diluted                           $0.31                  $0.23

Average number of common shares
outstanding (in thousands)
      Basic                             2,423                  2,366
      Diluted                           2,458                  2,406
</TABLE>

           See accompanying notes to condensed financial statements.

                                       3
<PAGE>
 
                        Kewaunee Scientific Corporation
                            Condensed Balance Sheets
                                ($ in thousands)

<TABLE>
<CAPTION>
                                                     July 31          April 30
                                                      1998              1998
                                                     -------          -------- 
Assets                                             (Unaudited)
------                                
<S>                                                <C>                <C>
Current assets:
  Cash and cash equivalents                             $471           $1,809
  Receivables                                         14,480           13,819
  Inventories                                          3,160            3,710
  Deferred income taxes                                1,240            1,240
  Prepaid expenses and other current assets              559              275
                                                     -------          ------- 
Total current assets                                  19,910           20,853
                                                     -------          ------- 
Property, plant and equipment, at cost                28,168           27,063
Accumulated depreciation                             (17,471)         (17,029)
                                                     -------          ------- 
Net property, plant and equipment                     10,697           10,034
                                                     -------          ------- 
Other assets                                           1,054              979
                                                     -------          ------- 
Total Assets                                         $31,661          $31,866
                                                     =======          =======

Liabilities and Stockholders' Equity
------------------------------------
Current liabilities:
  Accounts payable                                     6,715            6,209
  Employee compensation and amounts withheld           1,398            2,439
  Other current liabilities                            2,280            2,639
                                                     -------          ------- 
Total current liabilities                             10,393           11,287
                                                     -------          ------- 

Deferred income taxes                                    809              809
Accrued employee benefit plan costs                      733              731
                                                     -------          ------- 
Total Liabilities                                     11,935           12,827
                                                     -------          ------- 

Stockholders' equity:
  Common stock                                         6,550            6,550
  Additional paid-in-capital                             104              144
  Retained earnings                                   14,213           13,568
  Common stock in treasury, at cost                   (1,141)          (1,223)
                                                     -------          ------- 
Total stockholders' equity                            19,726           19,039
                                                     -------          ------- 
Total Liabilities and Stockholders' Equity           $31,661          $31,866
                                                     =======          =======
</TABLE>

           See accompanying notes to condensed financial statements.

                                       4
<PAGE>
 
                        Kewaunee Scientific Corporation
                       Condensed Statements of Cash Flows
                                  (Unaudited)
                                ($ in thousands)

<TABLE>
<CAPTION>
                                                           Three months ended
                                                                July 31
                                                          ---------------------
                                                           1998           1997
                                                          ------         ------
<S>                                                       <C>            <C> 
Cash flows from operating activities:
Net earnings                                                $766           $559
Adjustments to reconcile net earnings to net cash
used in operating activities:
  Depreciation and amortization                              456            394
  Provision for bad debts                                     49             40
  Increase in receivables                                   (710)        (1,693)
  (Increase) decrease in inventories                         550         (1,042)
Increase (decrease) in accounts payable and
    other current liabilities                               (894)           908
  Other, net                                                (358)          (182)
                                                          ------         ------
Net cash used in operating activities                       (141)        (1,016)
                                                          ------         ------

Cash flows from investing activities:
  Capital expenditures                                    (1,118)           (42)
                                                          ------         ------
Net cash used in investing activities                     (1,118)           (45)
                                                          ------         ------

Cash flows from financing activities:
  Net increase in short-term borrowings                       --          1,180
  Dividends paid                                            (121)           (95)
  Proceeds from exercise of stock options                     42             --
                                                          ------         ------
Net cash (used in) provided by financing activities          (79)         1,085
                                                          ------         ------

Increase (decrease) in cash and cash equivalents          (1,338)            24

Cash and cash equivalents, beginning of period             1,809              6
                                                          ------         ------

Cash and cash equivalents, end of period                    $471            $30
                                                          ======         ======
Supplemental disclosure:
  Interest paid                                              $10            $34
  Income taxes paid                                         $840           $466
</TABLE>

           See accompanying notes to condensed financial statements.

                                       5
<PAGE>
 
                        Kewaunee Scientific Corporation
                    Notes to Condensed Financial Statements
                                  (unaudited)

A.  Financial Information
-------------------------

The unaudited interim condensed financial statements of Kewaunee Scientific
Corporation (the "Company" or "Kewaunee") have been prepared pursuant to the
rules and regulations of the Securities and Exchange Commission (the
"Commission").  Accordingly, certain information and footnote disclosures
normally included in financial statements prepared in accordance with generally
accepted accounting principles have been condensed or omitted.  These interim
condensed financial statements should be read in conjunction with the financial
statements and notes included in the Company's 1998 Annual Report to
Stockholders.

The preparation of the financial statements in conformity with generally
accepted accounting principles requires management to make certain estimates and
assumptions that affect reported amounts and disclosures.  Actual results could
differ from those estimates.

In the opinion of management, the interim condensed financial statements reflect
all adjustments (consisting only of normal recurring adjustments) necessary for
a fair presentation of the interim periods.  The results of operations for the
interim periods are not necessarily indicative of the results of operations to
be expected for the full year.


B.  Inventories
---------------

Inventories consisted of the following (in thousands):

<TABLE>
<CAPTION>
                          July 31, 1998    April 30,1998
                          ------------------------------
<S>                       <C>              <C>
Finished products             $  522          $1,020
Work in process                1,120           1,106
Raw materials                  1,518           1,674
                              ------          ------
                              $3,160          $3,710
                              ======          ======
</TABLE>

C.  Balance Sheet
-----------------

The Company's April 30, 1998 condensed balance sheet as presented herein is
derived from audited financial statements, but does not include all disclosures
required by generally accepted accounting principles.

                                       6
<PAGE>
 
                 Item 2.  Management's Discussion and Analysis
                of Financial Condition and Results of Operations


The Company's 1998 Annual Report to Stockholders contains management's
discussion and analysis of financial condition and results of operations at and
for the year ended April 30, 1998.  The following discussion and analysis
describes material changes in the Company's financial condition since April 30,
1998.  The analysis of results of operations compares the three months ended
July 31, 1998 with the comparable period of the prior fiscal year.


Results of Operations
---------------------

The Company recorded sales of $19.6 million for the three months ended July 31,
1998, up 11.1% from sales of $17.7 million for the comparable period of the
prior year.  The sales increase for the current quarter resulted primarily from
increased unit sales of steel and wood laboratory furniture, partially offset by
lower sales of technical products.

The Company's gross profit margin for the three months ended July 31, 1998 was
22.5%, compared to 23.4% for the comparable period of the prior fiscal year.
The decrease in the gross profit margin for the quarter resulted primarily from
differences in the product sales mix for the two periods.

Operating expenses for the three months ended July 31, 1998 were $3.1 million,
down from operating expenses of $3.2 million for the comparable period of the
prior fiscal year.  As a percent of sales, operating expenses for the three
months ended July 31, 1998 were 15.9% of sales as compared to 17.9% of sales for
the comparable period of the prior fiscal year. The decrease in operating
expenses as a percent of sales for the quarter was primarily attributable to
significantly lower sales commissions paid.  The lower commissions resulted from
differences in the product sales mix for the two periods and an increase during
the current quarter of sales made directly to the Company's dealers for resale.

Operating earnings of $1.3 million were recorded for the three months ended July
31, 1998, as compared to $972,000 recorded in the comparable period of the prior
fiscal year.

                                       7
<PAGE>
 
Interest expense was $12,000 for the three months ended July 31, 1998, compared
to $52,000 for the comparable period of the prior fiscal year.  The decrease in
interest expense for the current quarter resulted primarily from lower levels of
debt under the Company's revolving credit facility.

Other expenses were $10,000 for the three months ended July 31, 1998, compared
to other income of $11,000 for the comparable period of the prior fiscal year.

Income tax expense of $511,000 was recorded for the three months ended July 31,
1998, as compared to an income tax expense of $372,000 recorded for the
comparable period of the prior fiscal year.  The effective tax rate was
approximately 40% for each of the periods.

Net earnings of $766,000, or $.31 per diluted share, was recorded for the three
months ended July 31, 1998, compared to net earnings of $559,000, or $.23 per
diluted share, for the comparable period of the prior fiscal year, as a result
of the factors discussed above.


Liquidity and Capital Resources
-------------------------------

Historically, the Company's principal sources of liquidity have been funds
generated from operations, supplemented as needed by short-term borrowings.  The
Company believes that these sources will be sufficient to support ongoing
business levels, including capital expenditures through the current fiscal year.

The Company had working capital of $9.5 million at July 31, 1998, as compared to
$9.6 million at April 30, 1998.  The ratio of current assets to current
liabilities was 1.9-to-1 at July 31, 1998, as compared to 1.8-to-1 at April 30,
1998.  The Company had no outstanding borrowings under its revolving credit
facility at July 31, 1998 and at April 30, 1998.

                                       8
<PAGE>
 
The Company's operations used cash of $141,000 during the three months ended
July 31, 1998.  This usage was primarily attributable to an increase in customer
receivables and a decrease in accounts payable, offset by operating earnings and
a decrease in inventory. The Company's operations used cash of $1.0 million
during the three months ended July 31, 1997, primarily for an increase in
customer receivables and inventories.

During the three months ended July 31, 1998, the Company used cash of $1.1
million for the purchase of production equipment, compared to the use of $45,000
for such purchases in the comparable period of the prior fiscal year.  In the
three months ended July 31, 1997, the Company entered into operating lease
arrangements for production equipment with an aggregate original asset cost of
$422,000.  The Company does not anticipate an abnormal level of capital
expenditures for the remainder of the current fiscal year.


Year 2000
---------

The Year 2000 issue is the result of computer programs being written using two
digits rather than four to define the applicable year.  Computer programs that
have date-sensitive software may recognize a date using "00" as the year 1900
rather than the year 2000.  In the event that the Company's internal systems or
one or more significant suppliers or customers fail to achieve Year 2000
compliance, the Company's business and profitability could be adversely
affected.

The Company has completed an inventory and preliminary assessment of its
computer systems and application software and has determined that its main
business applications are Year 2000 compliant.  Teams have been established to
implement plans to address Year 2000 issues for all other aspects of the
Company's business, including upgrading or replacing non-compliant software and
investigating the Year 2000 capabilities of suppliers, customers and other
external entities.  The Company does not yet have a formal contingency plan with
respect to suppliers, customers and other external entities, but will develop a
plan if determined necessary. A target date of April 30, 1999 has been set for
establishing compliance and developing any necessary contingency plans.  At this
time, the Company can not estimate a worst case Year 2000 scenario, but is
continuing to analyze the issue.

                                       9
<PAGE>
 
The Company has not incurred, nor does it expect to incur, significant costs in
addressing Year 2000 issues, as the Company's main business applications are
Year 2000 compliant.  Any such costs, including internal staff costs and costs
to write-off unamortized hardware and software that may need to be replaced,
will generally be expensed as incurred. The Company does not expect Year 2000
costs to materially affect its profitability or financial position.


Recent Accounting Standards
---------------------------

The Financial Accounting Standards Board has issued Statement of Financial
Accounting Standards (SFAS) No. 131, "Disclosures about Segments of an
Enterprise and Related Information."  This statement will be adopted as required
during the Company's 1999 fiscal year.  SFAS No. 131 redefines how operating
segments are determined and requires disclosure of certain financial and
descriptive information about a company's operating segments.  The Company has
not completed its evaluation of the effects that SFAS No. 131 will have on its
financial reporting and disclosures.


Safe Harbor Statement under the Private Securities Litigation Reform Act of 1995
--------------------------------------------------------------------------------

Certain statements in this report constitute "forward-looking" statements within
the meaning of the Private Securities Litigation Reform Act of 1995 (the "Reform
Act").  Such forward-looking statements involve known and unknown risks,
uncertainties and other factors that could significantly impact results or
achievements expressed or implied by such forward-looking statements.  These
factors include, but are not limited to, economic, competitive, governmental,
and technological factors affecting the Company's operations, markets, products,
services, and prices.  The cautionary statements made pursuant to the Reform Act
herein and elsewhere by the Company should not be construed as exhaustive or as
any admission regarding the adequacy of disclosures made by the Company prior to
the effective date of the Reform Act.  The Company cannot always predict what
factors would cause actual results to differ materially from those indicated by
the forward-looking statements.  In addition, readers are urged to consider
statements that include the terms "believes", "belief", "expects", "plans",
"objectives", "anticipates", "intends" or the like to be uncertain and forward-
looking.

                                       10
<PAGE>
 
                       REVIEW BY INDEPENDENT ACCOUNTANTS


A review of the interim financial information included in this Quarterly Report
on Form 10-Q for the three months ended July 31, 1998 has been performed by
PricewaterhouseCoopers LLP, the Company's independent accountants.  Their report
on the interim financial information follows.  There have been no adjustments or
disclosures proposed by PricewaterhouseCoopers LLP which have not been reflected
in the interim financial information.

                                       11
<PAGE>
 
REPORT OF INDEPENDENT ACCOUNTANTS


To the Board of Directors and Stockholders of
Kewaunee Scientific Corporation
Statesville, North Carolina

We have reviewed the accompanying condensed balance sheet of Kewaunee Scientific
Corporation as of July 31, 1998, and the related condensed statements of
operations and of cash flows for the three-month period then ended.  These
financial statements are the responsibility of the Company's management.

We conducted our review 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, the objective of which is
the expression of an opinion regarding the financial statements taken as a
whole.  Accordingly, we do not express such an opinion.

Based on our review, we are not aware of any material modifications that should
be made to the accompanying financial information for it to be in conformity
with generally accepted accounting principles.

We previously audited in accordance with generally accepted auditing standards,
the balance sheet as of April 30, 1998 and the related statements of income, of
retained earnings, and of cash flows for the year then ended (not presented
herein), and in our report dated June 1, 1998 we expressed an unqualified
opinion on those financial statements.  In our opinion, the information set
forth in the accompanying condensed balance sheet as of April 30, 1998, is
fairly stated in all material respects in relation to the balance sheet from
which it has been derived.



PricewaterhouseCoopers LLP
Charlotte, North Carolina

August 14, 1998

                                       12
<PAGE>
 
                          PART II.  OTHER INFORMATION


Item 4. Submission of Matters to a Vote of Security Holders


        The Company's Annual Meeting of Stockholders was held on August 26,
        1998.  Each of the nominees for Class III directors was re-elected.  The
        votes cast for and withheld from each such director were as follows:

<TABLE>
<CAPTION>
            Director                        For     Withheld
            --------                     ---------  --------
            <S>                          <C>        <C>
            Kingman Douglass             2,094,611    3,918
            Eli Manchester, Jr.          2,094,811    3,718
</TABLE>

Item 6. Exhibits and Reports on Form 8-K

        (a) Exhibits

            Exhibit 27    Financial Data Schedule

        (b) Reports on Form 8-K

            No reports on Form 8-K were filed with the Commission during the
            three months ended July 31, 1998.

                                       13
<PAGE>
 
                                   SIGNATURE


Pursuant to the requirements of the Securities and Exchange Act of 1934, the
registrant has duly caused this report to be signed on its behalf by the
undersigned thereunto duly authorized.



                                    KEWAUNEE SCIENTIFIC CORPORATION
                                    -------------------------------

                                              (Registrant)



Date:  September 9, 1998            By   /s/ D. Michael Parker
                                       ---------------------------
                                        D. Michael Parker
                                        Vice President of Finance
                                        Chief Financial Officer

                                       14