<PAGE>   1



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

                       SECURITIES AND EXCHANGE COMMISSION

                             WASHINGTON, D.C. 20549

                      ------------------------------------
                                    FORM 10-Q


(MARK ONE)

        [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

                          COMMISSION FILE NUMBER 1-9965



                           KEITHLEY INSTRUMENTS, INC.
             (Exact name of registrant as specified in its charter)

             OHIO                                    34-0794417
(State or other jurisdiction of                   (I.R.S. Employer
incorporation or organization)                   Identification No.)

                      28775 AURORA ROAD, SOLON, OHIO 44139
               (Address of principal executive offices) (Zip Code)

       REGISTRANT'S TELEPHONE NUMBER, INCLUDING AREA CODE: (440) 248-0400


         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 February 1, 1999 there were outstanding 4,852,421 Common Shares,
without par value, and 2,692,528 Class B Common Shares, without par value.

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

<PAGE>   2



                          PART I. FINANCIAL INFORMATION
                          -----------------------------

ITEM 1.  Financial Statements.
-------  ---------------------

                           KEITHLEY INSTRUMENTS, INC.
                           CONSOLIDATED BALANCE SHEET
                            (In Thousands of Dollars)
                                   (Unaudited)


<TABLE>
<CAPTION>
                                                           DECEMBER 31,             SEPTEMBER 30,
                                                           ------------             -------------
                                                      1998             1997             1998
                                                      ----             ----             ----
<S>                                                 <C>              <C>              <C>     
Assets
------

Current assets:
     Cash and cash equivalents                      $ 16,961         $  2,381         $  9,321
     Accounts receivable and other, net               13,785           18,681           17,586
     Inventories:
           Raw materials                               5,175            7,584            5,997
           Work in process                             2,667            6,352            3,163
           Finished products                           1,875            3,566            2,490
                                                    --------         --------         --------
             Total inventories                         9,717           17,502           11,650
     Other current assets                              3,873            3,490            3,770
                                                    --------         --------         --------
             Total current assets                     44,336           42,054           42,327
                                                    --------         --------         --------

Property, plant and equipment, at cost                39,211           42,368           39,334
Less-Accumulated depreciation                         25,526           25,001           24,723
                                                    --------         --------         --------
Net property, plant and equipment                     13,685           17,367           14,611
                                                    --------         --------         --------

Other assets                                          14,045           15,201           14,079
                                                    --------         --------         --------
Total assets                                        $ 72,066         $ 74,622         $ 71,017
                                                    ========         ========         ========


Liabilities and Shareholders' Equity
------------------------------------

Current liabilities:
     Short-term debt and current
       installments on long-term debt               $     --         $     --         $     --
     Accounts payable                                  6,207            9,202            6,191
     Accrued payroll and related expenses              4,490            4,173            4,203
     Other accrued expenses                            6,606            7,271            6,902
     Income taxes payable                              4,965            2,031            4,591
                                                    --------         --------         --------
           Total current liabilities                  22,268           22,677           21,887
                                                    --------         --------         --------

Long-term debt                                         6,599           14,337            6,099
Other long-term liabilities                            4,198            3,988            4,289

Shareholders' equity:
     Paid-in-capital                                   9,106            7,560            9,074
     Earnings reinvested in the business              33,418           26,674           29,870
     Accumulated other comprehensive income              375              198              429
     Unamortized portion of restricted stock            (272)            (525)            (283)
     Common shares held in treasury, at cost          (3,626)            (287)            (348)
                                                    --------         --------         --------
           Total shareholders' equity                 39,001           33,620           38,742
                                                    --------         --------         --------
Total liabilities and shareholders' equity          $ 72,066         $ 74,622         $ 71,017
                                                    ========         ========         ========
</TABLE>


                                       2

<PAGE>   3



                           KEITHLEY INSTRUMENTS, INC.
                        CONSOLIDATED STATEMENT OF INCOME
               (In Thousands of Dollars Except for Per Share Data)
                                   (Unaudited)


<TABLE>
<CAPTION>

                                                      FOR THE THREE MONTHS
                                                        ENDED DECEMBER 31,
                                                      1998              1997
                                                      ----              ----


<S>                                                 <C>              <C>     
Net sales                                           $ 20,881         $ 31,623

Cost of goods sold                                     8,806           13,039

Selling, general and administrative expenses           8,868           12,706

Product development expenses                           2,282            3,927

Gain on sale of business                              (4,808)              --

Net financing (income) expenses                          (39)             315
                                                    --------         --------

Income before income taxes                             5,772            1,636

Income taxes                                           1,989              540
                                                    --------         --------

Net income                                          $  3,783         $  1,096
                                                    ========         ========


Basic earnings per share                            $    .49         $    .14
                                                    ========         ========

Diluted earnings per share                          $    .48         $    .14
                                                    ========         ========

Cash dividends per Common Share                     $   .033         $   .031
                                                    ========         ========

Cash dividends per Class B
Common Share                                        $   .026         $   .025
                                                    ========         ========
</TABLE>



                                       3

<PAGE>   4



                           KEITHLEY INSTRUMENTS, INC.
                      CONSOLIDATED STATEMENT OF CASH FLOWS
                            (In Thousands of Dollars)
                                   (Unaudited)

<TABLE>
<CAPTION>
                                                                                        FOR THE THREE MONTHS
                                                                                         ENDED DECEMBER 31,
                                                                                      1998                1997
                                                                                      ----                ----
<S>                                                                                  <C>              <C>     
Cash flows from operating activities:
     Net income                                                                      $  3,783         $  1,096
     Expenses not requiring outlay of cash                                                735            1,158
     Gain on sale of business                                                          (4,808)              --
     Changes in working capital                                                         2,857            2,645
     Other operating activities                                                           (47)             174
                                                                                     --------         --------
     Net cash provided by operating activities                                          2,520            5,073
                                                                                     --------         --------

Cash flows from investing activities:
     Payments for property, plant, and equipment                                         (284)          (1,192)
     Sale of assets                                                                     8,947               --
     Payments made for sale of business                                                  (554)              --
     Other investing activities-net                                                         6               12
                                                                                     --------         --------
     Net cash provided by (used in) investing activities                                8,115           (1,180)
                                                                                     --------         --------

Cash flows from financing activities:
     Decrease in short-term debt                                                           --              (16)
     Borrowing (repayment) of long-term debt                                              500           (3,044)
     Cash dividends                                                                      (235)            (195)
     Repurchase of treasury stock                                                      (3,246)              --
     Other transactions-net                                                                --               44
                                                                                     --------         --------
     Net cash used in financing activities                                             (2,981)          (3,211)
                                                                                     --------         --------

Effect of changes in foreign currency exchange rates                                      (14)             (28)
                                                                                     --------         --------

Increase in cash and cash equivalents                                                   7,640              654
Cash and cash equivalents at beginning of period                                        9,321            1,727
                                                                                     --------         --------
Cash and cash equivalents at end of period                                           $ 16,961         $  2,381
                                                                                     ========         ========

SUPPLEMENTAL DISCLOSURES OF CASH FLOW INFORMATION 
     Cash paid during the period for:
           Income taxes                                                              $  1,591         $    245
           Interest                                                                        45              281

Disclosure of accounting policy
-------------------------------
     For purposes of this statement, the Company considers all highly liquid investments with maturities of three months
or less when purchased to be cash equivalents.
</TABLE>


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<PAGE>   5



                   NOTES TO CONSOLIDATED FINANCIAL STATEMENTS
                   ------------------------------------------
                (In thousands of dollars, except for share data)
                ------------------------------------------------


A.   The consolidated financial statements at December 31, 1998 and 1997, and
     for the three month periods then ended have not been examined by
     independents accountants, but in the opinion of the management of Keithley
     Instruments, Inc., all adjustments necessary to a fair statement of the
     consolidated balance sheet, consolidated statement of income and
     consolidated statement of cash flows for those periods have been included.
     All adjustments included are of a normal recurring nature.

B.   The weighted average number of shares and share equivalents used in
     determining basic earnings per share and diluted earnings per share was
     7,780,105 and 7,876,240 for the quarter ended December 31, 1998,
     respectively, and 7,667,227 and 7,994,582 for the quarter ended December
     31, 1997, respectively. Both Common Shares and Class B Common Shares are
     included in calculating the weighted average number of shares outstanding.

C.   On November 9, 1998, the company sold certain assets used in the operation
     of its Quantox product line to KLA-Tencor Corporation for $9,147 in cash,
     $8,947 of which was received upon closing with the remaining $200 placed in
     escrow. The agreement, which was effective October 31, 1998, included the
     sale of the Quantox inventory, certain machinery, equipment and other
     tangible personal property. The company retained the accounts receivable.
     The sale resulted in a pretax gain of $4,808, or $.39 per share after
     taxes.

D.    On November 11, 1998, the company commenced a tender offer to repurchase
     up to 2,000,000 of its Common Shares, or approximately 25 percent of the
     outstanding Common Shares and Class B Common Shares combined. The offer was
     conducted through a procedure commonly known as a "Dutch Auction" in which
     shareholders could tender their shares at prices not in excess of $7.00 nor
     less than $5.75 per share. The offer expired on December 10, 1998, and
     resulted in the repurchase of 405,733 Common Shares at $7.00 per share plus
     expenses of approximately $1.00 per share. At the conclusion of the Dutch
     Auction, the company's Board of Directors approved a program to repurchase
     up to 1,000,000 Common Shares, or approximately 13 percent of the
     outstanding Common Shares and Class B Common Shares combined, over a two-
     year period. The shares repurchased under both the Dutch Auction and the
     stock repurchase program will initially be held as treasury stock, and from
     time to time, may be reissued in settlement of stock options and the
     company's employee stock purchase plan. During the first quarter ended
     December 31, 1998, the company did not repurchase any shares other than
     those acquired through the Dutch Auction, and did not reissue any of the
     treasury shares.



                                       5

<PAGE>   6



E.   On October 1, 1998, the company adopted Statement of Financial Accounting
     Standards No. 130, "Reporting Comprehensive Income" (SFAS 130), which
     establishes rules for reporting comprehensive income and its components.
     The company's only item of other comprehensive income is foreign currency
     translation adjustments recorded in shareholders' equity. The adoption of
     SFAS 130 did not impact the company's net income or total shareholders'
     equity. Comprehensive income for the three months ended December 31, 1998
     and 1997 is as follows:

<TABLE>
<CAPTION>
                                                          December 31,
                                                      1998            1997
                                                      ----            ----

<S>                                                 <C>             <C>    
                Net income                          $ 3,783         $ 1,096
                Foreign currency translation
                  adjustments                           (54)            (52)
                                                    -------         -------

                Comprehensive income                $ 3,729         $ 1,044
                                                    =======         =======
</TABLE>



                                       6

<PAGE>   7


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

                            (In Thousands of Dollars)

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

First Quarter 1999 Compared with First Quarter 1998
---------------------------------------------------

Net income for the first quarter of fiscal 1999 was $3,783, or $.48 per share,
and included $3,090, or $.39 per share, for the gain on the sale of the
company's Quantox product line. Excluding the gain, net income was $693, or $.09
per share, compared with net income of $1,096, or $.14 per share, reported in
the first quarter of 1998.

Net sales of $20,881 decreased 34 percent from $31,623 in the prior year's first
quarter. Excluding sales from the Radiation Measurements Division (RMD) sold in
the fourth quarter of fiscal 1998 and the Quantox product line (Quantox) sold in
the first quarter of 1999, net sales were down 25 percent. This was principally
due to sales for the company's instrument products targeted for the wireless
communications industry, which were significantly below the strong sales in last
year's quarter. Geographically, sales from the company's current businesses
decreased in the United States and the Pacific Basin region, but increased
slightly in Europe. Orders for the first quarter of $24,231 decreased 30 percent
from the strong prior year's quarter. Excluding RMD and Quantox in both years,
orders were down 17 percent from the prior year's first quarter, but up
approximately 8 percent from the prior two quarters. Order backlog of $9,562 at
December 31, 1998, grew $3,528 during the quarter excluding Quantox backlog at
September 30, 1998.

Cost of goods sold as a percentage of net sales increased to 42.2 percent from
41.2 percent in the prior year's first quarter. This was mainly due to fixed
manufacturing costs being spread over lower sales volume. The effect of foreign
exchange hedging on cost of goods sold was immaterial in both periods.

Selling, general and administrative expenses decreased $3,838 from the prior
year's quarter, but increased as a percentage of net sales to 42.5 from 40.2 due
to lower sales volume. About two-thirds of the decrease is due to lower
marketing costs. Of total selling, general and administrative expenses, more
than one-third of the decrease can be attributed to zero costs in the current
year's quarter for RMD and only one month's worth of costs for Quantox. The
remainder of the decrease was due to continued cost control actions and last
year's workforce reductions.

Product development expenses for the quarter of $2,282, or 10.9 percent of net
sales, decreased from $3,927, or 12.4 percent of net sales, last year. About
two-thirds of the decrease was due to zero costs in the current year's quarter
for RMD and only one month's worth of costs for Quantox. The remainder of the
decrease was due to last year's workforce reductions.

On November 9, 1998, the company sold certain assets used in the operation of
its Quantox product line to KLA-Tencor Corporation. The sale was effective
October 31, 1998, and resulted in a gain of $4,808 pretax, or $.39 per share
after taxes. (See Note C.)

                                      7

<PAGE>   8

The company generated net financing income during the quarter of $39 versus
expense of $315 in the prior year. Lower average debt levels this year,
resulting in lower interest expense, combined with income earned on
significantly higher cash and cash equivalents accounted for the variance.

The effective tax rate for the quarter was 34.5 percent versus 33.0 percent last
year. The current quarter's higher rate was due to the sale of Quantox. The tax
rate on earnings excluding the gain was 28.1 percent, which was lower than the
statutory rate due to the utilization of tax credits.


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

During the first quarter, net cash provided by operations was $2,520 and cash
received from the sale of the Quantox product line was $8,947. The company used
available cash of $3,246 to repurchase 405,733 Common Shares through the Dutch
Auction tender offer conducted during the quarter. (See Note D.) Total cash and
cash equivalents increased $7,640 during the quarter to $16,961 at December 31,
1998. Total debt of $6,599 at December 31, 1998, increased $500 during the
quarter, and the total debt-to-capital ratio was 14.5 percent at December 31,
1998.

The company expects to finance debt service, capital spending, the stock
repurchase program and working capital requirements through cash on hand and
cash provided by operations. At December 31, 1998, the company had available
unused lines of credit with domestic and foreign banks aggregating $23,929 of
which $5,528 were short-term and $18,401 were long-term.


Outlook
-------

The company's management made the decision to sell RMD and Quantox not only for
strategic reasons, but also for financial reasons. These businesses were not
providing an adequate return on the company's investment as they were too far
from the company's core businesses and resources could not be efficiently
leveraged. Additionally, RMD and Quantox were less profitable than the core
businesses as measured by return on assets. As a result of the divestitures,
management has made significant changes to the cost structure that should allow
for earnings growth on incremental sales increases.

Due to the increase in backlog at December 31, 1998, management believes that 
sales and earnings for the second quarter of fiscal 1999 should exceed those 
reported for the first quarter, assuming second quarter orders are similar to 
those of the first quarter.


Factors That May Affect Future Results
--------------------------------------

Information included above in the Outlook section of Management's Discussion and
Analysis of Financial Condition and Results of Operations relating to
expectations as to the company's cost structure, earnings growth, financial
goals, sales and earnings for the second quarter and future financial results,
constitute "forward-looking" statements, as that term is defined in the Private
Securities Litigation Reform Act of 1995. Such statements are subject to certain
risks and uncertainties that could cause actual results to differ materially
from those projected. Some of the factors that may affect future results are
discussed below.



                                       8
<PAGE>   9


Although the company operates in a single industry segment, certain of its
products and product lines are sold into the semiconductor industry. Growth in
demand for semiconductors, new technology and pricing drive the demand for new
semiconductor capital equipment. Throughout much of the past two years, the
order growth of this industry has contracted which adversely affected revenues
of the company. If this trend continues, future revenues could also be adversely
affected.

The company's business relies on the development of new high technology products
and services to provide solutions to customer's complex measurement needs. This
requires anticipation of customers' changing needs and emerging technology
trends. The company must make long-term investments and commit significant
resources before knowing whether its expectations will eventually result in
products that achieve market acceptance. The company incurs significant expenses
developing new products that may or may not result in significant sources of
revenue and earnings in the future.

In many cases the company's products compete directly with those offered by
other manufacturers. If any of the company's competitors were to develop
products or services that are more cost-effective or technically superior,
demand for the company's product offerings could slow.

The company's cost structure is comprised of costs that are directly related to
the level of sales, as well as costs that are fixed and do not fluctuate based
on quarterly sales levels. The company's ability to maintain its cost structure
or to further improve its cost structure depends on its ability to control those
costs that are fixed or semi-variable.

The company currently has ten subsidiaries or sales offices located outside the
United States, and non-U.S. sales made up almost half of the company's revenue
in fiscal 1998. The company's future results could be adversely affected by
several factors, including the length and severity of the Asian financial
crisis, changes in foreign currency exchange rates, changes in a country's or
region's political or economic conditions, trade protection measures, import or
export licensing requirements, unexpected changes in regulatory requirements and
natural disasters.

The company recognizes the need to ensure that Year 2000 hardware and software
issues will not adversely impact its operations. With regard to the company's
own information systems, a substantial portion of Year 2000 information
technology compliance will be achieved in connection with the company's ongoing
program to upgrade its key information and operational systems. The company
believes that all key systems that are not already Year 2000 compliant will be
modified, upgraded or replaced prior to the year 2000, and that any related
costs will not have a material impact on the results of operations, financial
condition or cash flows of future periods. Certain of the company's hardware and
software products purchased by customers or currently being sold to customers
will require upgrade or other remediation to become Year 2000 compliant. Based
on an internal assessment of these products, the company does not believe that
the cost to modify these products for Year 2000 compliance will have a material
effect on the results of operations, financial condition or cash flows of future
periods. Lastly, the company is seeking to determine if the information systems
of its major suppliers (insofar as they relate to the company's business) comply
with Year 2000 requirements. The company has not yet fully determined the extent
to which its business may be impacted by third parties whose products and
services may not be ready for the year 2000. If it is determined that any third
party may not be 


                                      9

<PAGE>   10

ready, the company will develop a contingency plan. While management does not
expect that the failure of any third party to be fully compliant by 2000 would
significantly affect results of operations, financial condition or cash flows of
future periods, there can be no assurance that any such failure will not have an
adverse effect on the company's operations.

The company has modified its systems to accommodate the Euro. The cost of these
modifications was immaterial to the company's results of operations. Although
difficult to predict, any competitive implications and any impact on existing
financial instruments are expected to be immaterial to the company's results of
operations, financial condition or cash flows of future periods.


                                     10


<PAGE>   11


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


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

         (a) Exhibits. The following exhibits are filed herewith:
             ---------

           Exhibit
           Number          Exhibit
           ------          -------
           11    Statement Re Computation of Per Share Earnings

           27    Financial Data Schedule (EDGAR version only)


          (b)    During the first quarter ended December 31, 1998, the company
                 filed a Current Report on Form 8-K under Item 2 - Acquisition
                 or Disposition of Assets, for the sale of certain assets of its
                 Quantox product line. The Form 8-K, dated November 9, 1998,
                 included an Unaudited Pro forma Consolidated Balance Sheet as
                 of June 30, 1998, Unaudited Pro forma Consolidated Statements
                 of Income for the fiscal year ended September 30, 1997 and the
                 nine months ended June 30, 1998, and Notes thereto.


                                   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.


                             KEITHLEY INSTRUMENTS, INC.
                             (Registrant)






Date: February 12, 1999      /s/ Joseph P. Keithley
                             -------------------------
                                 Joseph P. Keithley
                                 Chairman, President and Chief Executive Officer
                                 (Principal Executive Officer)






Date: February 12, 1999      /s/ Mark J. Plush
                             --------------------
                                 Mark J. Plush
                                 Vice President and Chief Financial Officer
                                 (Principal Financial and Accounting Officer)




                                       11