<PAGE>   1
                       SECURITIES AND EXCHANGE COMMISSION

                             WASHINGTON, D.C. 20549

                                    FORM 10-Q



          [X] QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE
                         SECURITIES EXCHANGE ACT OF 1934


                  For the Quarterly Period Ended March 31, 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 0-13546

                               ------------------


                     APACHE OFFSHORE INVESTMENT PARTNERSHIP
             ------------------------------------------------------
             (Exact Name of Registrant as Specified in Its Charter)


                Delaware                                         41-1464066
    -------------------------------                       ----------------------
    (State or Other Jurisdiction of                          (I.R.S. Employer
     Incorporation or Organization)                       Identification Number)
                                                                                


    Suite 100, One Post Oak Central                             77056-4400
  2000 Post Oak Boulevard, Houston, TX                          ----------
----------------------------------------                        (Zip Code)
(Address of Principal Executive Offices)


       Registrant's Telephone Number, Including Area Code: (713) 296-6000



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
                                -----          -----






<PAGE>   2
                         PART I - FINANCIAL INFORMATION


ITEM 1 - FINANCIAL STATEMENTS


                     APACHE OFFSHORE INVESTMENT PARTNERSHIP
                               STATEMENT OF INCOME
                                   (UNAUDITED)


<TABLE>
<CAPTION>
                                                  FOR THE THREE MONTHS
                                                     ENDED MARCH 31,
                                                -------------------------
                                                   1998           1997
                                                ----------     ----------
<S>                                             <C>            <C>       
REVENUES:
   Oil and gas sales                            $2,195,685     $3,698,691
   Interest income                                   9,811          9,392
                                                ----------     ----------

                                                 2,205,496      3,708,083
                                                ----------     ----------

EXPENSES:
   Depreciation, depletion and amortization        722,475        841,467
   Lease operating                                 205,656         91,168
   Administrative                                  135,000        135,000
   Interest                                             --         12,818
                                                ----------     ----------

                                                 1,063,131      1,080,453
                                                ----------     ----------

NET INCOME                                      $1,142,365     $2,627,630
                                                ==========     ==========

NET INCOME ALLOCATED TO:
   Managing Partner                             $  317,545     $  611,456
   Investing Partners                              824,820      2,016,174
                                                ----------     ----------

                                                $1,142,365     $2,627,630
                                                ==========     ==========

NET INCOME PER INVESTING PARTNER UNIT           $      697     $    1,683
                                                ==========     ==========


WEIGHTED AVERAGE INVESTING PARTNER
   UNITS OUTSTANDING                               1,184.2        1,197.9
                                                ==========     ==========
</TABLE>




                 The accompanying notes to financial statements
                     are an integral part of this statement.


                                       1
<PAGE>   3



                     APACHE OFFSHORE INVESTMENT PARTNERSHIP
                             STATEMENT OF CASH FLOWS
                                   (UNAUDITED)


<TABLE>
<CAPTION>
                                                                    FOR THE THREE MONTHS
                                                                       ENDED MARCH 31,
                                                                ----------------------------
                                                                   1998              1997
                                                                -----------      -----------
<S>                                                             <C>              <C>        
CASH FLOWS FROM OPERATING ACTIVITIES:
   Net income                                                   $ 1,142,365      $ 2,627,630
   Adjustments to reconcile net income to net cash
      provided by operating activities:
         Depreciation, depletion and amortization                   722,475          841,467
         Changes in operating assets and liabilities:
             Decrease in accrued revenues receivable                114,253        1,069,507
             Decrease in accrued operating expenses payable         (90,825)         (52,874)
             (Increase) decrease in receivable from
                 Apache Corporation                                   1,185       (1,382,853)
                                                                -----------      -----------

         Net cash provided by operating activities                1,889,453        3,102,877
                                                                -----------      -----------

CASH FLOWS FROM INVESTING ACTIVITIES:
   Additions to oil and gas properties                           (1,427,745)        (738,219)
   Non-cash portion of oil and gas property additions              (196,202)         235,372
   Proceeds from sales of oil and gas properties                    363,534               --
   Decrease in drilling advances                                     13,141               --
                                                                -----------      -----------

         Net cash used in investing activities                   (1,247,272)        (502,847)
                                                                -----------      -----------

CASH FLOWS FROM FINANCING ACTIVITIES:
   Distributions to Investing Partners                                   --       (1,197,827)
   Distributions to Managing Partner, net                          (273,370)        (690,627)
   Payments on long-term debt                                            --       (1,997,500)
                                                                -----------      -----------

         Net cash used in financing activities                     (273,370)      (3,885,954)
                                                                -----------      -----------

NET INCREASE (DECREASE) IN CASH AND
   CASH EQUIVALENTS                                                 368,811       (1,285,924)

CASH AND CASH EQUIVALENTS, BEGINNING OF YEAR                        691,797        1,737,470
                                                                -----------      -----------

CASH AND CASH EQUIVALENTS, END OF PERIOD                        $ 1,060,608      $   451,546
                                                                ===========      ===========

SUPPLEMENTAL DISCLOSURE OF CASH FLOW
   INFORMATION:
   Cash paid during the period for interest                     $        --      $    11,073
                                                                ===========      ===========
</TABLE>


                 The accompanying notes to financial statements
                     are an integral part of this statement.


                                       2
<PAGE>   4



                     APACHE OFFSHORE INVESTMENT PARTNERSHIP
                                 BALANCE SHEET
                                  (Unaudited)
<TABLE>
<CAPTION>
                                                                                              MARCH 31,        DECEMBER 31,
                                                                                                 1998               1997
                                                                                            -------------      -------------
<S>                                                                                         <C>                <C>          
                                  ASSETS

CURRENT ASSETS:
   Cash and cash equivalents                                                                $   1,060,608      $     691,797
   Accrued revenues receivable                                                                  1,423,198          1,537,451
   Receivable from Apache Corporation                                                             217,544            218,729
   Drilling advances                                                                               58,879             72,020
                                                                                            -------------      -------------

                                                                                                2,760,229          2,519,997
                                                                                            -------------      -------------

OIL AND GAS PROPERTIES, on the basis of full cost accounting:
      Proved properties                                                                       167,461,298        166,397,087
      Less - Accumulated depreciation, depletion
         and amortization                                                                    (159,115,389)      (158,392,914)
                                                                                            -------------      -------------

                                                                                                8,345,909          8,004,173
                                                                                            -------------      -------------

                                                                                            $  11,106,138      $  10,524,170
                                                                                            =============      =============

                    LIABILITIES AND PARTNERS' CAPITAL

CURRENT LIABILITIES:
   Accrued exploration and development                                                      $   1,568,715      $   1,764,917
   Accrued operating expenses payable and other                                                    21,438            112,263
                                                                                            -------------      -------------

                                                                                                1,590,153          1,877,180
                                                                                            -------------      -------------

PARTNERS' CAPITAL:
   Managing Partner                                                                               549,210            505,035
   Investing Partners (1,184.2 and 1,184.2 units
      outstanding, respectively)                                                                8,966,775          8,141,955
                                                                                            -------------      -------------

                                                                                                9,515,985          8,646,990
                                                                                            -------------      -------------

                                                                                            $  11,106,138      $  10,524,170
                                                                                            =============      =============
</TABLE>

                 The accompanying notes to financial statements
                     are an integral part of this statement.


                                       3
<PAGE>   5



                     APACHE OFFSHORE INVESTMENT PARTNERSHIP
                          NOTES TO FINANCIAL STATEMENTS
                                   (UNAUDITED)


     The financial statements included herein have been prepared by the Apache
Offshore Investment Partnership (the Partnership), without audit, pursuant to
the rules and regulations of the Securities and Exchange Commission, and reflect
all adjustments which are, in the opinion of management, necessary for a fair
statement of the results for the interim periods, on a basis consistent with the
annual audited financial statements. All such adjustments are of a normal,
recurring nature. Certain information, accounting policies, and footnote
disclosures normally included in financial statements prepared in accordance
with generally accepted accounting principles have been omitted pursuant to such
rules and regulations, although the Partnership believes that the disclosures
are adequate to make the information presented not misleading. These financial
statements should be read in conjunction with the financial statements and the
summary of significant accounting policies and notes thereto included in the
Partnership's latest annual report on Form 10-K.


1.   OTHER ACCRUED EXPENSES

     Accrued expenses payable at March 31, 1998, primarily represented operating
costs accrued in February and March that will be paid in subsequent months.

2.   RECEIVABLE FROM APACHE

     The receivable from Apache Corporation, the Partnership's managing partner
(Apache or the Managing Partner), represents the net result of the Investing
Partners' revenue and expenditure transactions in the current month. Generally,
cash in this amount will be transferred from Apache in the month after the
Partnership's transactions are processed and the net results of operations are
determined.

3.   RIGHT OF PRESENTMENT

     In February 1994, an amendment to the Partnership Agreement created a right
of presentment under which all Investing Partners have a limited and voluntary
right to offer their Units to the Partnership twice each year to be purchased
for cash. The first right of presentment offer for 1998 was based upon a
valuation date of December 31, 1997 for a purchase price of $11,161 per Unit,
plus interest to the date of payment. The offer was made to the Investing
Partners on April 28, 1998, and Investing Partners may elect to exercise their
right of presentment through May 29, 1998.

     The Partnership is not in a position to predict how many Units will be
presented for repurchase under the April 1998 offer and cannot, at this time,
determine if the Partnership will have sufficient funds available to repurchase
Units. The Partnership has no obligation to purchase any Units presented to the
extent it determines that it has insufficient funds for such purchases. The
amended Partnership Agreement contains limitations on the number of Units that
the Partnership can repurchase, including a limit of 10 percent of the
outstanding Units on an annual basis.


                                       4
<PAGE>   6



ITEM 2 - MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND
         RESULTS OF OPERATIONS

RESULTS OF OPERATIONS

NET INCOME AND REVENUE

     The Partnership reported net income of $1.1 million in the first quarter of
1998, versus $2.6 million in the prior year period. Net income per Investing
Partner Unit decreased 59 percent, from $1,683 per Unit to $697 per Unit. The
decrease was attributable to lower natural gas production and prices, lower
crude oil prices and higher lease operating expense (LOE), mitigated by lower
depreciation, depletion and amortization (DD&A) expense and financing costs.

     Revenues decreased 41 percent, from $3.7 million in the first quarter of
1997 to $2.2 million for the same period in 1998. Natural gas and crude oil
sales contributed 81 percent and 18 percent, respectively, to the Partnership's
total revenues, with one percent attributable to interest income.

     The Partnership's oil and gas production volume and price information is
summarized in the following table (gas volumes presented in thousand cubic feet
(Mcf) per day):

<TABLE>
<CAPTION>
                                                     FOR THE THREE MONTHS
                                                        ENDED MARCH 31,
                                               ----------------------------------
                                                   1998                 1997                DECREASE
                                               -------------        -------------         --------------
<S>                                                   <C>                 <C>                     <C>  
          Gas volume - Mcf per day                    9,079               11,596                  (22%)
          Average gas price - per Mcf                 $2.19                $2.83                  (23%)
          Oil volume - barrels per day                  333                  393                  (15%)
          Average oil price - per barrel             $13.50               $21.15                  (36%)
</TABLE>

FIRST QUARTER 1998 COMPARED TO FIRST QUARTER 1997

     Natural gas sales revenues for the first quarter of 1998 totaled $1.8
million, 40 percent lower than the first quarter of 1997. The decrease resulted
from a 22 percent decline in natural gas production, negatively impacting
revenue by $.5 million. Natural gas production decreased as a result of natural
declines in production, the sale of High Island Block A-6 (Glenda Prospect) in
October 1997, and the sale of West Cameron Block 368 (Krypton Prospect) in
January 1998. Also contributing to the decrease in revenues was a 23 percent
decrease in natural gas prices, negatively impacting revenue by $.7 million.

     The Partnership's crude oil sales revenues for the first quarter totaled
$.4 million, a 46 percent decrease from the first quarter of 1997. The impact of
a 36 percent decrease in average realized oil prices negatively impacted
revenues by $.3 million. The decrease in production was primarily the result of
natural declines.

OPERATING EXPENSES

     The Partnership's DD&A expense for the first quarter of 1998 decreased 14
percent from the same period in the previous year due to lower oil and gas sales
revenue. The Partnership's DD&A rate, expressed as a percentage of sales, was 33
percent during the first three months of 1998, increasing from 23 percent during
the same period in 1997. The increase in the rate was a result of generally
declining natural gas and crude oil prices and downward reserve revisions
recorded in the fourth quarter of 1997.



                                       5
<PAGE>   7


     LOE in the first quarter of 1998 increased from $.1 million to $.2 million
compared to the first quarter of 1997. The increase was the result of higher
workover activity in the first three months of 1998 and joint venture audit
credits received in March 1997.

CASH FLOW, LIQUIDITY AND CAPITAL RESOURCES

CAPITAL RESOURCES AND LIQUIDITY

     The Partnership's primary capital resource is net cash provided by
operating activities, which was $1.9 million for the first three months of 1998,
a decrease of 39 percent from a year ago, driven by lower oil and gas production
and prices. Future cash flows will similarly be influenced by fluctuations in
product prices, production levels and operating costs.

     The Partnership repaid all outstanding debt and terminated its revolving
credit facility on January 31, 1997, rendering the Partnership debt free for the
first time in its 14-year history, and has not incurred any new debt in 1998.

     It is expected that the net cash provided by operating activities will be
sufficient to meet the Partnership's liquidity needs through the end of 1998.
However, in the event short-term operating cash requirements are greater than
the Partnership's financial resources, the Partnership will seek short-term,
interest-bearing advances from the Managing Partner.

CAPITAL COMMITMENTS

     The Partnership's primary needs for cash are for operating expenses,
drilling and recompletion expenditures, distributions to Investing Partners and
the purchase of Units offered by Investing Partners under the right of
presentment.

     During the first three months of 1998, the Partnership's oil and gas
property additions totaled $1.4 million. These additions related primarily to
completions at South Timbalier 295 and drilling activity at South Pass 83. Based
on information supplied by the operators of the properties, the Partnership
anticipates capital expenditures of approximately $3.0 million for the remainder
of 1998. The anticipated capital expenditures relate to completions and
additional workovers at South Timbalier 295 and the remaining drilling activity
at South Pass 83. Such estimates may change based on realized prices, drilling
results or changes by the operator to the development plan.

     As provided in the amended Partnership Agreement, a first right of
presentment offer for 1998 of $11,161 per Unit, plus interest to the date of
payment, was made to Investing Partners on April 28, 1998, based on a valuation
date of December 31, 1997. The Partnership is not in a position to predict how
many Units will be presented for repurchase during 1998 and cannot, at this
time, determine if the Partnership will have sufficient funds available to
repurchase Units. The Partnership has no obligation to purchase any Units
presented to the extent it determines that it has insufficient funds for such
purchases.

     Due primarily to the reduced level of cash provided by operating
activities, and the expected need to utilize such cash to fund anticipated
capital expenditures, the Partnership did not make a distribution in the first
quarter of 1998. The Managing Partner will periodically reassess the
Partnership's cash position to determine if a distribution later in the year
will be possible. The amount of future distributions will be dependent on actual
and expected production levels, oil and gas prices, and drilling and
recompletion expenditures.



                                       6
<PAGE>   8



IMPACT OF THE YEAR 2000 ISSUE

      The "Year 2000 Issue" is the result of computer software being written
using two digits rather than four to define the applicable year. Apache, as
Managing Partner, manages the Partnership's operations. Apache uses a portion of
its staff for this purpose and is reimbursed for actual costs paid on behalf of
the Partnership, as well as for general, administrative, and overhead costs
properly allocable to the Partnership. Any of the Managing Partner's computer
programs that have date-sensitive software may recognize a date using "00" as
the year 1900 rather than the year 2000. If left unremedied, this could result
in a system failure or miscalculations causing disruptions of operations
including, among other things, a temporary inability to process transactions or
engage in similar normal business activities.

      The Managing Partner is in the process of replacing significant portions
of its software to more effectively and efficiently meet its business needs.
Replacement computer systems selected by Apache will properly recognize dates
beyond December 31, 1999. The Managing Partner presently believes that with
conversions to new software, the Year 2000 Issue will be eliminated. However, if
such conversions are not made, or are not completed timely, the Year 2000 Issue
could have a material impact on the operations of the Partnership. The Managing
Partner plans to replace substantially all of its existing systems within 12
months or not later than March 31, 1999.

      The date on which the Managing Partner plans to complete installation of
its new system is based on management's best estimates, which were derived using
numerous assumptions of future events including the continued availability of
certain resources. However, there can be no guarantee that these estimates will
be achieved and actual results could differ materially from those plans.
Specific factors that might cause such material differences include, but are not
limited to, the availability and cost of personnel trained in this area and
similar uncertainties.

FORWARD-LOOKING STATEMENTS AND RISK

     Certain statements in this report, including statements of the future
plans, objectives, and expected performance of the Partnership, are
forward-looking statements that are dependent on certain events, risks and
uncertainties that may be outside the Partnership's control which could cause
actual results to differ materially from those anticipated. Some of these
include, but are not limited to, economic and competitive conditions, inflation
rates, legislative and regulatory changes, financial market conditions,
political and economic uncertainties of foreign governments, future business
decisions, and other uncertainties, all of which are difficult to predict.

     There are numerous uncertainties inherent in estimating quantities of
proved oil and gas reserves and in projecting future rates of production and
timing of development expenditures. The total amount or timing of actual future
production may vary significantly from reserves and production estimates. The
drilling of exploratory wells can involve significant risks including those
related to timing, success rates and cost overruns. Lease and rig availability,
complex geology and other factors can affect these risks. Future oil and gas
prices also could affect results of operations and cash flows.



                                       7
<PAGE>   9



                           PART II - OTHER INFORMATION


ITEM 1.      LEGAL PROCEEDINGS

             None.

ITEM 2.      CHANGES IN SECURITIES

             None.

ITEM 3.      DEFAULTS UPON SENIOR SECURITIES

             None.

ITEM 4.      SUBMISSION OF MATTERS TO A VOTE OF SECURITY HOLDERS

             None.

ITEM 5.      OTHER INFORMATION

             None.

ITEM 6.      EXHIBITS AND REPORTS ON FORM 8-K

             a.   Exhibits.

                  27.1     Financial Data Schedule.

             b.   Reports on Form 8-K - None.



                                       8
<PAGE>   10




                                   SIGNATURES


Pursuant to the requirements of Section 13 or 15(d) 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.


                                         APACHE OFFSHORE INVESTMENT PARTNERSHIP
                                         By: Apache Corporation, General Partner



Dated:    May 12, 1998                   /s/ Roger B. Plank
                                         ---------------------------------------
                                         Roger B. Plank
                                         Vice President and Chief Financial 
                                         Officer


Dated:    May 12, 1998                   /s/ Thomas L. Mitchell
                                         ---------------------------------------
                                         Thomas L. Mitchell
                                         Vice President and Controller
                                         (Chief Accounting Officer)

<PAGE>   11
                                 Exhibit Index

<TABLE>
<CAPTION>
Exhibit
Number         Description
-------        -----------
  <S>          <C>
  27.1         Financial Data Schedule
</TABLE>