<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 September 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 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 QUARTER          FOR THE NINE MONTHS
                                                        ENDED SEPTEMBER 30,        ENDED SEPTEMBER 30,
                                                   ---------------------------   --------------------------
                                                       1998           1997           1998          1997
                                                   ------------   ------------   ------------   ------------
<S>                                                <C>            <C>            <C>            <C>         
REVENUES:
   Oil and gas production revenues                 $  1,713,277   $  2,751,530   $  6,062,749   $  9,150,173
   Interest income                                       21,820         41,214         47,068         66,376
   Other income                                              --         89,250             --         89,250
                                                   ------------   ------------   ------------   ------------
                                                      1,735,097      2,881,994      6,109,817      9,305,799
                                                   ------------   ------------   ------------   ------------
EXPENSES:
   Depreciation, depletion and amortization             622,671        693,231      2,070,659      2,167,320
   Lease operating                                      380,910         83,130        771,372        340,606
   Administrative                                       129,137        135,000        405,116        405,000
   Interest                                                  --             --             --         12,818
                                                   ------------   ------------   ------------   ------------
                                                      1,132,718        911,361      3,247,147      2,925,744
                                                   ------------   ------------   ------------   ------------
NET INCOME                                         $    602,379   $  1,970,633   $  2,862,670   $  6,380,055
                                                   ============   ============   ============   ============
NET INCOME ALLOCATED TO:
   Managing Partner                                $    195,817   $    462,058   $    826,414   $  1,493,315
   Investing Partners                                   406,562      1,508,575      2,036,256      4,886,740
                                                   ------------   ------------   ------------   ------------

                                                   $    602,379   $  1,970,633   $  2,862,670   $  6,380,055
                                                   ============   ============   ============   ============

NET INCOME PER INVESTING PARTNER UNIT              $        352   $      1,268   $      1,737   $      4,090
                                                   ============   ============   ============   ============

WEIGHTED AVERAGE INVESTING PARTNER
   UNITS OUTSTANDING                                    1,154.4        1,190.1        1,172.5        1,194.8
                                                   ============   ============   ============   ============
</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 NINE MONTHS ENDED 
                                                                                       SEPTEMBER 30,
                                                                              --------------------------
                                                                                  1998           1997
                                                                              -----------    -----------
<S>                                                                           <C>            <C>        
CASH FLOWS FROM OPERATING ACTIVITIES:
   Net income                                                                 $ 2,862,670    $ 6,380,055
   Adjustments to reconcile net income to net cash
      provided by operating activities:
         Depreciation, depletion and amortization                               2,070,659      2,167,320
         Changes in operating assets and liabilities:
             Decrease in accrued revenues receivable                              344,054      1,975,005
             Increase (decrease) in accrued operating expenses payable             77,852       (193,482)
             Decrease in payable to/receivable from
                 Apache Corporation                                               135,274       (548,380)
                                                                              -----------    -----------
         Net cash provided by operating activities                              5,490,509      9,780,518
                                                                              -----------    -----------
CASH FLOWS FROM INVESTING ACTIVITIES:
   Additions to oil and gas properties                                         (2,144,221)    (2,322,219)
   Non-cash portion of oil and gas property additions                            (951,355)       200,123
   Proceeds from sales of oil and gas properties                                  363,534             --
   (Increase) decrease in drilling advances                                        72,020        (37,326)
                                                                              -----------    -----------
         Net cash used in investing activities                                 (2,660,022)    (2,159,422)
                                                                              -----------    -----------

CASH FLOWS FROM FINANCING ACTIVITIES:
   Repurchase of Partnership Units                                               (348,000)      (114,132)
   Distributions to Investing Partners                                                 --     (1,197,827)
   Distributions to Managing Partner, net                                        (804,419)    (1,833,177)
   Payments on long-term debt                                                          --     (1,997,500)
                                                                              -----------    -----------
         Net cash used in financing activities                                 (1,152,419)    (5,142,636)
                                                                              -----------    -----------

NET INCREASE IN CASH AND CASH EQUIVALENTS                                       1,678,068      2,478,460

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

CASH AND CASH EQUIVALENTS, END OF PERIOD                                      $ 2,369,865    $ 4,215,930
                                                                              ===========    ===========
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

<TABLE>
<CAPTION>
                                                                      SEPTEMBER 30,      DECEMBER 31,
                                                                          1998              1997
                                                                    -----------------  -----------------
                                                                     (Unaudited)
                                     ASSETS
<S>                                                                 <C>                <C>            
CURRENT ASSETS:
   Cash and cash equivalents                                        $     2,369,865    $       691,797
   Accrued revenues receivable                                            1,193,397          1,537,451
   Receivable from Apache Corporation                                        83,455            218,729
   Drilling advances                                                             --             72,020
                                                                    ---------------    ---------------
                                                                          3,646,717          2,519,997
                                                                    ---------------    ---------------

OIL AND GAS PROPERTIES, on the basis of full cost accounting:
   Proved properties                                                    168,177,774        166,397,087
   Less - Accumulated depreciation, depletion
         and amortization                                              (160,463,573)      (158,392,914)
                                                                    ---------------    ---------------
                                                                          7,714,201          8,004,173
                                                                    ---------------    ---------------

                                                                    $    11,360,918    $    10,524,170
                                                                    ===============    ===============

                       LIABILITIES AND PARTNERS' CAPITAL

CURRENT LIABILITIES:
   Distribution payable                                             $     1,731,539    $            --
   Accrued exploration and development                                      813,562          1,764,917
   Accrued operating expenses payable and other                             190,115            112,263
                                                                    ---------------    ---------------
                                                                          2,735,216          1,877,180
                                                                    ---------------    ---------------

PARTNERS' CAPITAL:
   Managing Partner                                                         527,030            505,035
   Investing Partners (1,154.4 and 1,184.2 units
      outstanding, respectively)                                          8,098,672          8,141,955
                                                                    ---------------    ---------------
                                                                          8,625,702          8,646,990
                                                                    ---------------    ---------------
                                                                    $    11,360,918    $    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.    PAYABLE TO/RECEIVABLE FROM APACHE

     Payable to/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 to/from Apache in the month
after the Partnership's transactions are processed and the net results of
operations are determined.


2.    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, as a result, the Partnership acquired 29.832
Units for $348,000 in cash. A second right of presentment offer of $9,824 per
Unit, plus interest to the date of payment, was made to the Investing Partners
on October 30, 1998, based on a valuation date of June 30, 1998.

     The Partnership is not in a position to predict how many Units will be
presented for repurchase during the fourth quarter of 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. The Partnership Agreement, as amended (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 $.6 million in the third quarter of
1998, versus $2.0 million in the prior year period. Net income per Investing
Partner Unit decreased 72 percent, from $1,268 per Unit to $352 per Unit. The
decrease was primarily attributable to lower natural gas and crude oil
production and prices and higher lease operating expense (LOE).

     For the first nine months of 1998, net income of $2.9 million, or $1,737
per Investing Partner Unit, decreased 55 percent and 58 percent, respectively,
from $6.4 million, or $4,090 per Investing Partner Unit, in the same period
last year. Lower natural gas and crude oil production and prices and higher LOE
impacted 1998 results.

     Revenues decreased 40 percent, from $2.9 million in the third quarter of
1997 to $1.7 million for the same period in 1998. Natural gas and crude oil
sales contributed 80 percent and 19 percent, respectively, to the Partnership's
total revenues in the third quarter of 1998. For the first nine months of 1998,
revenues decreased 34 percent to $6.1 million compared to $9.3 million for the
same period in 1997, primarily due to lower production, with natural gas and
crude oil contributing 81 percent and 18 percent, respectively, to the
Partnership's total revenues in 1998.

     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 QUARTER ENDED         FOR THE NINE MONTHS ENDED
                                                    SEPTEMBER 30,                     SEPTEMBER 30,
                                      --------------------------------------------------------------------------
                                          1998        1997      DECREASE     1998          199        DECREASE
                                      ----------   ----------   --------- ----------   ----------   ------------
<S>                                   <C>          <C>          <C>       <C>          <C>          <C>  
Gas volume - Mcf per day                   7,857       10,726    (27%)         8,622       11,172     (23%)
Average gas price - per Mcf           $     1.91   $     2.25    (15%)    $     2.11   $     2.39     (12%)
Oil volume - barrels per day                 287          313     (8%)           305          349     (13%)
Average oil price - per barrel        $    12.60   $    18.43    (32%)    $    13.10   $    19.53     (33%)
</TABLE>


THIRD QUARTER 1998 COMPARED TO THIRD QUARTER 1997

     Natural gas sales revenues for the third quarter of 1998 totaled $1.4
million, 38 percent lower than the third quarter of 1997. The decrease resulted
from a 27 percent decline in natural gas production. Production volumes
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. Natural gas prices
decreased 15 percent for the third quarter of 1998 compared to the same period
in 1997, which negatively impacted revenue by $.3 million.

     The Partnership's crude oil sales revenues for the third quarter of 1998
totaled $.3 million, a 37 percent decrease from the third quarter of 1997. A 32
percent decrease in the average realized oil price led to the decline.


                                       5
<PAGE>   7



YEAR-TO-DATE 1998 COMPARED TO YEAR-TO-DATE 1997

     Natural gas sales for the first nine months of 1998 of $5.0 million
decreased $2.3 million, or 32 percent, when compared to the same period in
1997. Average realized gas prices decreased $.28 per Mcf, or 12 percent, when
compared with the first nine months of 1997. Natural gas production for the
first nine months of 1998 decreased by 23 percent when compared to the same
period in 1997, negatively impacting revenues by $1.5 million. Production
decreases in 1998 were primarily due to natural declines in production and the
sale of High Island Block A-6 and West Cameron Block 368.

     For the nine months ended September 30, 1998, oil sales decreased 41
percent to $1.1 million when compared to the same period last year. The
Partnership's oil sales revenues were negatively impacted by a 13 percent
decline in oil production and a 33 percent decrease in realized prices. The
decrease in production volumes resulted from natural declines in production.


OPERATING EXPENSES

     The Partnership's depreciation, depletion and amortization (DD&A) rate,
expressed as a percentage of oil and gas production revenues, was approximately
36 percent and 34 percent during the third quarter and first nine months of
1998, increasing from 25 percent and 24 percent, respectively, during the same
periods in 1997. The increase in the DD&A rate was a result of generally
declining natural gas and crude oil prices and downward reserve revisions
recorded in the fourth quarter of 1997.

     Higher workover activity in the third quarter of 1998 caused LOE to be 358
percent higher when compared to the third quarter of 1997. For the first nine
months of 1998, LOE of $.8 million was up 126 percent when compared to the
first nine months of 1997. This variance was the result of higher workover
activity in 1998 and joint venture audit refunds 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 $5.5 million for the first nine months of 1998,
a decrease of 44 percent from a year ago, driven by lower oil and gas
production and prices. Future cash flows will be similarly 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. The Partnership has not incurred any new
debt in 1998.

     It is expected that 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.


                                       6
<PAGE>   8


     During the first nine months of 1998, the Partnership's oil and gas
property additions totaled $2.1 million. These additions related primarily to
completions at South Timbalier Block 295, drilling activity at South Pass Block
83 and recompletions and drilling at Ship Shoal Block 259. Based on information
supplied by the operators of the properties, the Partnership anticipates
capital expenditures of approximately $.5 million for the remainder of 1998.
The anticipated capital expenditures relate to the drilling of two new wells at
North Padre Block 989. Such estimates may change based on realized prices,
drilling results or changes by the operator to the development plan.

     The Partnership made a distribution of $1,500 per Unit on October 1, 1998.
The amount of future distributions will be dependent on actual and expected
production levels, realized and expected oil and gas prices, and expected
drilling and recompletion expenditures.

     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. As a result, the Partnership acquired 29.832 Units
for a total of $348,000 in cash. A second right of presentment offer for 1998
of $9,824 per Unit, plus interest to the date of payment, was made to Investing
Partners on October 30, 1998, based on a valuation date of June 30, 1998. The
Partnership is not in a position to predict how many Units will be presented
for repurchase during the fourth quarter of 1998 and cannot, at this time,
determine if the Partnership will have sufficient funds available to repurchase
all Units presented.


                                       7
<PAGE>   9


IMPACT OF THE YEAR 2000 ISSUE

     The Year 2000 Issue poses a serious threat of business disruption to any
organization that utilizes computer technology and computer chip technology in
their business systems or equipment. 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. Apache has formed a Year 2000 Task Force with representation from
major business units to inventory and assess the risk of hardware, software,
telecommunications systems, office equipment, embedded chip controls and
systems, process control systems, facility control systems and dependencies on
external trading partners. The project phases, expected completion dates and
percentage complete are as follows:

<TABLE>
<CAPTION>

            PHASE                           COMPLETION DATE    % COMPLETE
---------------------------------------     ---------------   ------------
<S>                                         <C>               <C> 
Organization                                    July 1998        100%

Assessment                                  November 1998         70%
    Desktop Computers
    Network Hardware
    Software
    Embedded Systems
    External Trading Partners
    Building/Infrastructure Systems
    Telecommunications Systems

Implementation/Replacement                      July 1999         40%
    Computer Hardware    
    Core Business Software
    Desktop Software
    Embedded Systems
    Building Systems

Contact External Trading Partners           February 1999         10%

Contingency Planning                           March 1999         20%
</TABLE>


     To date, the Managing Partner is not aware of any significant issues that
would cause problems in the area of safety, environmental or business
interruption in the Year 2000. Apache will assess the risk associated with
hardware, software, infrastructure, embedded chips and external trading
partners that are not Year 2000 compliant. While Apache is confident that Year
2000 remediation efforts will succeed in minimizing exposure to business
disruption, plans are being developed which will allow continuation of business
in all but the worst case scenarios. All remediation and replacement efforts
and contingency planning are expected to be complete by July 1999. All critical
external trading partners will be contacted to determine Year 2000 readiness
and contingency plans will be developed where assurance of Year 2000 compliance
is not received by February 28, 1999.

     In 1997, the Managing Partner initiated a project to replace existing
business software as it relates to Apache's production, land, marketing,
accounting and financial systems to more effectively and efficiently meet its
business needs. Replacement computer systems selected by the Managing Partner
from SAP America, Inc., PricewaterhouseCoopers LLP, Innovative Business
Solutions and Landmark Graphics will properly recognize dates beyond December
31, 1999. The Managing Partner plans to implement the replacement software by
March 31, 1999. The business system replacement project is 60 percent complete
and Apache believes that the March 31, 1999 deadline is attainable.


                                       8
<PAGE>   10


     The Managing Partner expects that its cost to achieve Year 2000 compliance
will not exceed $4 million. These costs will be borne by the Managing Partner
and will not have any impact on the financial results of the Partnership. The
cost of implementing business replacement systems is not included in these cost
estimates.

     The Managing Partner presently believes that with conversions to new
software and completion of efforts planned by the Year 2000 Task Force, the
risk associated with Year 2000 will be significantly reduced. However, the
Managing Partner is unable to assure that the consequences of Year 2000
failures of systems maintained by the Managing Partner or by third parties will
not materially adversely impact the Partnership's results of operations,
liquidity or financial condition.


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 and 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.


                                       9
<PAGE>   11


                          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.


                                      10
<PAGE>   12


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


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

<PAGE>   13
                                    EXHIBITS



<TABLE>
<CAPTION>
EXHIBIT
NUMBERS        DESCRIPTION
--------   -------------------------------------------------
<S>        <C>
  27.1     Financial Data Schedule.
</TABLE>