<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 JUNE 30, 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 August 6, 1998 the Registrant had outstanding 5,084,135 Common
Shares, without par value, and 2,785,378 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>
                                                                             JUNE 30,                     SEPTEMBER 30,
                                                                      ------------------------            -------------
                                                                       1998              1997                  1997
                                                                       ----              ----                  ----
Assets
------

<S>                                                                   <C>              <C>                    <C>    
Current assets:
     Cash and cash equivalents                                        $ 2,412          $  2,018               $ 1,727
     Accounts receivable and other, net                                18,155            21,038                25,113
     Inventories:
           Raw materials                                                7,512             7,585                 7,787
           Work in process                                              4,198             5,508                 5,671
           Finished products                                            3,912             3,800                 3,121
                                                                      -------           -------               -------
             Total inventories                                         15,622            16,893                16,579
     Other current assets                                               3,206             3,016                 3,107
                                                                      -------           -------               -------
             Total current assets                                      39,395            42,965                46,526
                                                                       ------            ------                ------

Property, plant and equipment, at cost                                 42,825            41,822                41,527
Less-Accumulated depreciation                                          26,187            24,896                24,272
                                                                       ------            ------                ------
Total property, plant and equipment, net                               16,638            16,926                17,255
                                                                       ------            ------                ------

Other assets                                                           15,368            14,589                15,332
                                                                       ------            ------                ------
Total assets                                                          $71,401           $74,480               $79,113
                                                                       ======            ======                ======


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

Current liabilities:
     Short-term debt and current
       installments on long-term debt                               $      --          $     31             $      16
     Accounts payable                                                   6,357             9,089                11,568
     Accrued payroll and related expenses                               4,815             4,141                 4,698
     Other accrued expenses                                             5,985             7,336                 6,951
     Income taxes payable                                               2,106             1,244                 1,821
                                                                      -------           -------               -------
           Total current liabilities                                   19,263            21,841                25,054
                                                                       ------            ------                ------

Long-term debt                                                         12,265            17,865                17,442
Other long-term liabilities                                             3,928             3,531                 3,934

Shareholders' equity:
     Paid-in-capital                                                    8,867             7,076                 7,489
     Earnings reinvested in the business                               27,636            24,787                25,773
     Cumulative translation adjustment                                    205               218                   250
     Unamortized portion of restricted stock                             (436)             (625)                 (569)
     Common shares held in treasury, at cost                             (327)             (213)                 (260)
                                                                     --------          --------              --------
           Total shareholders' equity                                  35,945            31,243                32,683
                                                                       ------            ------                ------
Total liabilities and shareholders' equity                            $71,401           $74,480               $79,113
                                                                       ======            ======                ======
</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            FOR THE NINE MONTHS
                                                          ENDED JUNE 30,                 ENDED JUNE 30,
                                                      1998            1997            1998            1997
                                                      ----            ----            ----            ----

<S>                                                 <C>             <C>             <C>             <C>     
Net sales                                           $ 28,578        $ 32,410        $ 89,897        $ 88,444

Cost of goods sold                                    12,135          13,896          38,182          37,396

Selling, general and administrative expenses          12,111          12,898          36,499          37,287

Product development expenses                           3,088           3,878          10,271          12,752

Special charges                                           --             306             335             739

Net financing expenses                                   248             344             816             817
                                                    --------        --------        --------        --------


Income (loss) before income taxes                        996           1,088           3,794            (547)

Income taxes (benefit)                                   329             327           1,252            (132)
                                                    --------        --------        --------        --------


Net income (loss)                                   $    667        $    761        $  2,542        $   (415)
                                                    ========        ========        ========        ========


Basic income (loss) per share                       $   0.08        $   0.10        $   0.33        $  (0.05)
                                                    ========        ========        ========        ========

Diluted income (loss) per share                     $   0.08        $   0.10        $   0.32        $  (0.05)
                                                    ========        ========        ========        ========

Cash dividends per Common Share                     $  0.031        $  0.031        $  0.094        $  0.094
                                                    ========        ========        ========        ========

Cash dividends per Class B
    Common Share                                    $  0.025        $  0.025        $  0.075        $  0.075
                                                    ========        ========        ========        ========
</TABLE>


                                       3
<PAGE>   4


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

<TABLE>
<CAPTION>
                                                          FOR THE THREE MONTHS            FOR THE NINE MONTHS
                                                              ENDED JUNE 30,                  ENDED JUNE 30,
                                                          1998             1997           1998             1997
                                                          ----             ----           ----             ----
<S>                                                      <C>             <C>             <C>             <C>     
Cash flows from operating activities:
     Net income (loss)                                   $   667         $   761         $ 2,542         $  (415)
     Expenses not requiring outlay of cash                   944             732           3,197           3,188
     Changes in working capital                            1,007            (290)          1,931          (4,095)
     Other operating activities                             (595)         (1,278)           (200)           (477)
                                                         -------         -------         -------         -------
     Net cash provided by (used in)
       operating activities                                2,023             (75)          7,470          (1,799)
                                                         -------         -------         -------         -------

Cash flows from investing activities:
     Payments for property, plant, and equipment            (503)           (828)         (2,349)         (4,958)
     Other investing activities-net                           19             103              92             119
                                                         -------         -------         -------         -------
     Net cash used in investing activities                  (484)           (725)         (2,257)         (4,839)
                                                         -------         -------         -------         -------

Cash flows from financing activities:
     Net decrease in short term debt                          --             (15)            (16)            (30)
     Net borrowing (repayment) of long term debt          (1,041)          1,303          (5,085)          4,920
     Cash dividends                                         (229)           (222)           (679)           (662)
     Other transactions-net                                  104               4           1,311             778
                                                         -------         -------         -------         -------
     Net cash provided by (used in)
       financing activities                               (1,166)          1,070          (4,469)          5,006
                                                         -------         -------         -------         -------

Effect of exchange rate changes on cash                        6              11             (59)           (345)
                                                         -------         -------         -------         -------

Increase (decrease) in cash and cash equivalents             379             281             685          (1,977)
Cash and cash equivalents at beginning of period           2,033           1,737           1,727           3,995
                                                         -------         -------         -------         -------
Cash and cash equivalents at end of period               $ 2,412         $ 2,018         $ 2,412         $ 2,018
                                                         =======         =======         =======         =======

Supplemental disclosures of cash flow information
-------------------------------------------------
     Cash paid during the period for:
           Income taxes                                  $   344         $   399         $   855         $ 1,706
           Interest                                          238             306             748             819
</TABLE>

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.


                                       4
<PAGE>   5


                   NOTES TO CONSOLIDATED FINANCIAL STATEMENTS
                   ------------------------------------------

A.   The consolidated financial statements at June 30, 1998 and 1997 and for the
     three month periods then ended have not been examined by independent
     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 net income per share and diluted net income per share was
     7,856,302 and 8,041,919 for the quarter ended June 30, 1998, respectively,
     and 7,635,763 and 7,877,808 for the quarter ended June 30, 1997,
     respectively. The weighted average number of shares and share equivalents
     used to determine basic net income per share and diluted net income per
     share was 7,776,752 and 8,028,510 for the nine months ended June 30, 1998,
     respectively, and 7,570,062 for the nine months ended June 30, 1997. Both
     Common Shares and Class B Common Shares are included in calculating the
     weighted average number of shares outstanding.

C.   On August 10, 1998, the company sold certain assets used in the operation
     of its Radiation Measurements Division ("RMD") to Inovision Radiation
     Measurements, L.L.C. ("Inovision") pursuant to an Asset Purchase Agreement
     entered into and effective as of July 31, 1998. The major asset categories
     include RMD's inventory, accounts receivable, machinery, equipment,
     furniture and other tangible personal property, and intangible assets
     including patents, technology and goodwill. The purchase price for the
     transaction was $8,215,000 in cash which was paid in full upon closing.
     Additionally, Inovision has assumed certain liabilities of RMD including
     accrued vacation, real estate taxes, accrued consultants, warranty, certain
     contracts and commissions, at an amount not to exceed $735,000.

D.   During the second quarter of fiscal 1998, the company incurred special
     charges of $335 relating to cost reductions announced in February 1998 for
     the company's semiconductor business, net of $10 for the reversal of
     personnel related expenses previously accrued for the Keithley MetraByte
     operation relocation. Special charges of $306 for the third quarter of
     fiscal 1997 include $706 for the relocation of the Keithley MetraByte
     operation to Cleveland from Taunton, Massachusetts and a reversal of $400
     of expense recorded during 1996 for accrued lease costs. Special charges of
     $739 for the fiscal 1997 nine month period include $1,339 for the
     relocation of the Keithley MetraByte operation to Cleveland from Taunton,
     Massachusetts and a reversal of $600 of expense recorded during 1996 for
     accrued lease costs. $224 of these special charges were non-cash. At June
     30, 1998 and 1997, the company had accruals of $658 and $2,026 remaining on
     the Consolidated Balance Sheet, respectively, ($108 and $1,685 classified
     as current, respectively, and $550 and $341, respectively, classified as
     long-term). The accruals represent costs related to the closing of the
     Taunton facility, personnel and related expenses due to cost reduction
     actions and certain European operating subleases.

E.   During the first quarter of fiscal 1997, $846 of income tax related
     accruals were released to income as circumstances giving rise to the
     accruals were no longer present. Also during that quarter, and in response
     to a recent tax law change, the company decided to terminate existing




                                       5
<PAGE>   6


     corporate owned life insurance policies and a deferred tax charge of $888
     was recorded, reflecting the tax on the excess of cash surrender value over
     premiums paid on the policies. The tax is payable over 4 years.












                                       6
<PAGE>   7


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

                            (In Thousands of Dollars)

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

Third Quarter 1998 Compared with Third Quarter 1997
---------------------------------------------------

Net income for the third quarter of fiscal 1998 was $667, or $0.08 per share,
compared with $761, or $0.10 per share, in the prior year's third quarter. Net
income for the 1998 third quarter includes expenses of $662 pretax, or $0.06 per
share after taxes, principally from personnel cost reduction actions. The 1997
third quarter's earnings include special charges of $306 pretax, or $0.02 per
share after taxes, for the relocation of a business. The company's new
businesses were essentially at breakeven for the third quarter of fiscal 1998,
compared to a loss of $2,646 in the prior year's quarter; however, earnings from
the company's traditional businesses were down substantially from the prior
year. The slowdown in the semiconductor industry and Asian financial situation
had a significant impact on the company's business.

Net sales of $28,578 decreased 12 percent from $32,410 in the prior year's third
quarter. Sales declined in all businesses, including the semiconductor business
unit where an increase in sales for the company's Quantox(R) product did not
offset the larger decline in sales of parametric test systems. Orders for the
third quarter were $27,927, down 22 percent from record order levels reported in
the prior year's quarter. Geographically, orders were down significantly in the
U.S., down in the Pacific Basin region, but up in Europe. The prior year
included unusually large orders for the company's instrument products targeted
for the telecommunications industry. Additionally, orders for the company's
parametric test systems used by the semiconductor industry were down
significantly. Order backlog decreased $1.2 million for the quarter to $13.7
million at June 30, 1998.

Cost of goods sold as a percentage of net sales decreased 0.4 percentage points
to 42.5 percent from 42.9 percent. The company's hedging activities had no
effect on cost of goods sold for the quarter versus a decrease of 0.2 percentage
points in the prior year's quarter.

Selling, general and administrative expenses of $12,111 for the third quarter
decreased 6 percent from $12,898 in the prior year's quarter, but due to lower
sales, increased to 42.3 percent of sales compared to 39.8 percent last year.
Lower commissions due to the lower sales as well as a different geographic
channel mix accounted for the majority of the decrease. The 1998 quarter's
expense also includes $662 principally for personnel reduction actions taken
during the quarter. Before these charges, selling, general and administrative
expenses would have been 40.1 percent of sales.

Product development expenses of $3,088, or 10.8 percent of net sales, decreased
20 percent from $3,878, or 12.0 percent of net sales, in the prior year's
quarter. The decrease was due to completion of several of the company's major
project initiatives including Quantox and the company's SmartLink(TM) line of
products. Additionally, the company implemented cost reduction actions in
February 1998 affecting the semiconductor business unit.

Special charges of $306 for the third quarter of fiscal 1997 include the
relocation of the Keithley MetraByte operation to Cleveland from Taunton,
Massachusetts and a reversal of expense recorded during 1996 for accrued lease
costs. $224 of special charges were non-cash. See Note D.


                                       7
<PAGE>   8



Net financing expenses of $248 for the third quarter of fiscal 1998 decreased 28
percent from $344 in last year's quarter, due to lower average debt levels
during the quarter.


Nine Months Ended June 30, 1998 Compared with Nine Months Ended June 30, 1997
-----------------------------------------------------------------------------

Net income for the nine months ending June 30, 1998, was $2,542, or $.32 per
share on a diluted basis, compared to a net loss of $415, or $.05 per share for
the nine month period last year. The earnings turnaround was principally due to
significantly lower losses from the company's new businesses.

Net sales of $89,897 increased 2 percent from $88,444 reported for the nine
month period last year. The increase was due mainly to increased shipments of
Quantox and the company's instrument products, as well as increased billing to
the Navy, mostly offset by lower sales of the company's parametric test
equipment and board products. Orders for the nine month period were down 8
percent from last year. Geographically, orders were down in the United States
and Pacific Basin region, but up in Europe.

Cost of goods sold as a percentage of net sales increased slightly to 42.5
percent from 42.2 percent for the nine month period last year. The company's
hedging activities had no effect on cost of goods versus a decrease of 0.1
percentage points for the nine months last year.

Selling, general and administrative expenses of $36,499 decreased $788, or 2
percent, from $37,287 in the same period last year.

Product development expenses of $10,271, or 11.4 percent of net sales, decreased
19 percent from $12,752, or 14.4 percent of net sales, in the prior year. The
decrease was due to completion of several of the company's major project
initiatives including Quantox and the S600 parametric tester, which were notable
expenses last year, and the company's SmartLink(TM) line of products.
Additionally, the company implemented cost reduction actions in February 1998
affecting the semiconductor business unit.

During the second quarter of fiscal 1998, the company incurred special charges
of $335 relating to cost reductions announced in February 1998 for the company's
semiconductor business, net of $10 for the reversal of personnel related
expenses previously accrued for the Keithley MetraByte operation relocation.
Special charges of $739 for the nine month period during fiscal 1997 include
$1,339 for the relocation of the Keithley MetraByte operation to Cleveland from
Taunton, Massachusetts and a reversal of $600 of expense recorded during 1996
for accrued lease costs. $224 of special charges recorded in the 1997 period
were non-cash. At June 30, 1998, the company had $658 remaining on the
Consolidated Balance Sheet ($108 in the caption "Other accrued expenses" and
$550 in the caption "Other long-term liabilities") related to the special
charges.

Net financing expenses of $816 for the nine months were flat versus $817 in the
prior year. Lower interest expense due to lower average debt levels during the
nine month period was offset by lower interest income due primarily to interest
income received last year on a federal income tax refund.


                                       8
<PAGE>   9



The company's effective tax rate was 33.0 percent for the nine-month period
compared with 24.1 percent last year. During the prior year's nine month period,
$846 of income tax related accruals were released to income as circumstances
giving rise to the accruals were no longer present. Also during last year's
period, and in response to a tax law change, the company decided to terminate
existing corporate owned life insurance policies and a deferred tax charge of
$888 was recorded, reflecting the tax on the excess of cash surrender value over
premiums paid on the policies. The tax is payable over 4 years.


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

Cash generated from operations was $2,023 for the third quarter and $7,470 for
the nine months ended June 30, 1998, and was used principally to pay down
long-term debt and purchase equipment. Additionally, financing activities for
the nine month period include $1,311 generated from the issuance of company
stock through the company's employee stock purchase plan and the exercise of
employee stock options. Total debt of $12,265 at June 30, 1998, decreased $981
during the quarter and $5,193 from the beginning of the fiscal year. The total
debt-to-capital ratio was 25.4 percent at June 30, 1998 compared with 34.8
percent at September 30, 1997.

The company expects to finance capital spending and working capital requirements
with cash provided by operations. At June 30, 1998, the Company had available
unused lines of credit with domestic and foreign banks aggregating $18,119, of
which $5,384 were short term and $12,735 were long term.


Outlook
-------

Two of the industries on which the company focuses, namely semiconductor and
electronic components, are experiencing very weak market conditions. The
continuing deterioration of these markets during the third quarter, resulting in
part from the ongoing Asian financial situation and DRAM over capacity, cause
management to be cautious over the near-term. Management now believes that
earnings for the fourth quarter will likely be lower than those of the third
quarter before the expenses of $0.06 per share for personnel cost reduction
actions. This would result in second half earnings being flat to somewhat down
from the first half of $0.27 per share, excluding special charges and the
one-time gain discussed below. Management is continuing to adjust the company's
cost structure, while focusing efforts on those businesses believed to have the
greatest growth and earnings potential.

On August 10, 1998, the company sold certain assets (net of assumed liabilities)
used in the operation of its Radiation Measurements Division pursuant to an
Asset Purchase Agreement entered into and effective as of July 31, 1998. As a
result of the sale, the company expects to recognize a one-time gain of $0.18 to
$0.21 per share in its fiscal fourth quarter ending September 30, 1998.


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

Certain information included above in the last paragraph in the Liquidity and
Capital Resources section and the Outlook section of Management's Discussion and
Analysis of Financial Condition and 



                                       9
<PAGE>   10


Results of Operations relating to expectations of cash flows and earnings,
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.

Although the company operates in a single industry segment, certain of its
products and product lines including the Quantox unit and the company's line of
parametric test systems, are sold to the semiconductor industry. Growth in
demand for semiconductors and new technology drives the demand for new
semiconductor capital equipment. Historically, sales and orders levels for this
business have been volatile and contractions in order levels can adversely
affect revenue for the company.

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 predictions will eventually result in
products that achieve market acceptance.

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 currently has ten subsidiaries or sales offices located outside the
United States, and non-U.S. sales made up 48 percent of total revenue for the
first nine months of fiscal 1998. The company's future results could be
adversely affected by several factors, including the length and severity of the
contraction in the semiconductor industry, 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 financial, 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. There
can be no assurance that the systems of other companies which the company deals
with will be able to adequately address the Year 2000 issue, or that the failure
to do so will not have an adverse effect on the company's operations.


                                       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)       REPORTS ON FORM 8-K. No reports on Form 8-K were filed during the 
             quarterly period ended June 30, 1998.



                                   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:  August 13, 1998            /s/  Joseph P. Keithley
                                 ----------------------------------------------
                                 Joseph P. Keithley
                                 Chairman, President and Chief Executive Officer
                                   (Principal Executive Officer)






Date:  August 13, 1998           /s/  Ronald M. Rebner
                                 ----------------------------------------------
                                 Ronald M. Rebner
                                 Vice President and Chief Financial Officer
                                 (Principal Financial and Accounting Officer)



                                       11