<PAGE>   1
                                 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 June 30, 1998

                                       OR

[ ] 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   1-5542

                               DEXTER CORPORATION
             (Exact name of registrant as specified in its charter)

              CONNECTICUT                              06-0321410
     (State or other jurisdiction of                (I.R.S. Employer
      incorporation or organization)               Identification No.)

ONE ELM STREET, WINDSOR LOCKS, CONNECTICUT               06096
 (Address of principal executive offices)              (Zip Code)

                                (860) 292-7675
              (Registrant's telephone number, including area code)

 N/A
(Former name, former address and former fiscal year, if changed since last
report.)

Indicate by check mark whether the registrant (1) has filed all reports required
to be filed by Section 13 or 15(d) of the Securities Exchange Act of 1934 during
the preceding 12 months (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 / /

Indicate the number of shares outstanding of each of the issuer's classes of
common stock, as of the latest practicable date.


<TABLE>
<CAPTION>
        CLASS                                       Outstanding at July 31, 1998
<S>                                                 <C>
COMMON STOCK, PAR VALUE $1                                23,033,879 SHARES
</TABLE>
<PAGE>   2
                                     PART I

                              FINANCIAL INFORMATION

Item 1 -     Financial Statements

             Reference is made to the following consolidated financial
             statements which are incorporated herein by reference:

              (a) Exhibit 99a - Condensed Statement of Income for the three and
                  six-month periods ended June 30, 1998 and 1997.

              (b) Exhibit 99b - Condensed Statement of Financial Position as of
                  June 30, 1998, December 31, 1997, and June 30, 1997.

              (c) Exhibit 99c - Condensed Statement of Cash Flows for the
                  six-month periods ended June 30, 1998 and 1997.

              (d) Exhibit 99d - Statement of Comprehensive Income for the three
                  and six-month periods ended June 30, 1998 and 1997.

              (e) Exhibit 99e - Net Sales by Market for the three and six-month
                  periods ended June 30, 1998 and 1997.

              (f) Exhibit 99f - Notes to Consolidated Financial Statements.

             The unaudited financial data included herein as of June 30, 1998
             and 1997, and for the three and six-month periods then ended, have
             been reviewed by the registrant's independent public accountants,
             PricewaterhouseCoopers LLP, and their report is attached.


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

Analysis of Operations

The Company reported record second quarter 1998 net income of $16.4 million, or
$.70 per share on a diluted basis, compared with $16 million, or $.69 per share
diluted for the second quarter of 1997. Net income increased 3% while both
diluted and basic earnings per share increased 1% compared with the second
quarter of 1997.

Sales for the second quarter of 1998 were a record $302.6 million, a 3% increase
compared with sales of $293.2 million in the second quarter of 1997. A 4%
increase in unit volume and a 2% increase due to acquisitions were partially
offset by a 2% unfavorable effect of currency translation rates and price
decreases averaging 1%.
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Item 2 -     Management's Discussion and Analysis of Financial
             Condition and Results of Operations, continued

Analysis of Operations, continued

Sales for the six-month period ended June 30, 1998 increased $27 million, or 5%,
to a record $592.5 million compared with sales of $565.5 million in 1997. A 5%
increase in unit volume and a 3% increase due to acquisitions were partially
offset by a 2% unfavorable effect of currency translation rates and price
decreases averaging 1%.

Earnings for the first six months of 1998 were a record $30.6 million, or $1.31
per share on a diluted basis, compared with $28.9 million, or $1.23 per share
diluted, for the first six months of 1997. This represents a 6% increase in net
income and basic earnings per share and a 7% increase in diluted earnings per
share.

Products with strong performance in the second quarter and first six months of
1998 include aerospace adhesives and coatings, food and specialty can coatings
serving international markets, and sales of products at Life Technologies, Inc.
Sales of nonwoven materials serving the food packaging market were stronger in
the second quarter of 1998 compared with the second quarter of 1997, and sales
of magnetic materials were stronger for the first six months of 1998 compared
with last year.

Sales of electronic encapsulation materials were weaker in the second quarter of
1998 compared with the second quarter of 1997. Additionally, printed wiring
board products serving the electronics market had weaker performance in both the
second quarter and first six months of 1998. Sales of beer and beverage can
coatings serving international markets showed improvement over the first quarter
of 1998 but were still below 1997 levels.

Consolidated gross margin of 36.6% in the second quarter of 1998, stated as a
percentage of sales, increased .5 percentage points compared with 36.1% for the
second quarter of 1997. Gross margin of 36.5% for the first six months of 1998
increased .6 percentage points compared with 35.9% for the same period last
year. These improvements came from wholly owned Dexter operations, primarily due
to volume increases and productivity improvements.

Marketing and administrative costs increased $2.9 million, or 5%, in the second
quarter of 1998 and $7.9 million, or 7%, for the first six months of 1998
compared with the same periods in 1997. These increases were principally due to
increased costs at Life Technologies, Inc. and marketing and administrative
costs associated with businesses acquired in the fourth quarter of 1997.

Other income for the first six months of 1998 of $4.7 million decreased $2.3
million, or 33%, compared with the first six months of 1997, primarily due to
lower equity income resulting from the divestiture of D & S Plastics
International, which was effective April 1, 1997, and lower interest income. The
impact of lower other income for the first six months of 1998 was partially
offset by lower interest expense and a lower effective tax rate of 35% in 1998
compared with 36% in 1997.
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Item 2 -     Management's Discussion and Analysis of Financial
             Condition and Results of Operations, continued

Analysis of Financial Condition

Excess acquisition cost as of June 30, 1998 was $93.8 million, an increase of
$17.5 million, compared with $76.3 million as of June 30, 1997. This increase
was primarily due to an increase of $24.7 million related to the impact of
businesses acquired in the fourth quarter of 1997, which was partially offset by
$6.6 million of amortized costs.

Other assets as of June 30, 1998 were $74.4 million, an increase of $24.5
million, compared with $49.9 million as of June 30, 1997. This increase was
primarily due to an increase of $23.1 million for patents, technology, formulas,
and covenants related to businesses acquired in the fourth quarter of 1997.
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                                     PART II

                                OTHER INFORMATION


Item 6  -   Exhibits and Reports on Form 8-K

        (a) Exhibit 15 of Part 1 - Letter to Securities and Exchange Commission
            re: Incorporation of Accountants' Report

            Exhibit 27 of Part 1 - Financial Data Schedule

            Exhibit 99 of Part 1 - Second Quarter 1998 Financial Statements and
            Notes

        (b) No reports on Form 8-K were filed during the quarter for which
            this report was filed.
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                                   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.



                                              DEXTER CORPORATION
                                              (Registrant)

Date        August 13, 1998                   /s/ Kathleen Burdett
            ---------------                   -----------------------------
                                              Kathleen Burdett
                                              Vice President and
                                              Chief Financial Officer
                                              (Principal Financial Officer)





Date        August 13, 1998                   /s/ Glenn E. Tynan
            ---------------                   -----------------------------
                                              Glenn E. Tynan
                                              Controller
                                              (Principal Accounting Officer)
<PAGE>   7
                                INDEX TO EXHIBITS


     Exhibit No.

         15       Letter to Securities and Exchange Commission re: Incorporation
                  of Accountants' Report

         27       Financial Data Schedule

         99       Second Quarter 1998 Financial Statements and Notes