<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, 1999

                                       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,
                                                    ----------------------------
                                                       1999              1998
                                                    ----------        ----------

<S>                                                 <C>               <C>       
 REVENUES:
    Oil and gas production revenues                 $1,663,778        $2,195,685
    Interest income                                     16,350             9,811
                                                    ----------        ----------
                                                     1,680,128         2,205,496
                                                    ----------        ----------

 EXPENSES:
    Depreciation, depletion and amortization           673,620           722,475
    Lease operating expense                            153,865           205,656
    Administrative                                     135,000           135,000
                                                    ----------        ----------
                                                       962,485         1,063,131
                                                    ----------        ----------
 NET INCOME                                         $  717,643        $1,142,365
                                                    ==========        ==========

 NET INCOME ALLOCATED TO:
    Managing Partner                                $  231,761        $  317,545
    Investing Partners                                 485,882           824,820
                                                    ----------        ----------
                                                    $  717,643        $1,142,365
                                                    ==========        ==========
 NET INCOME PER INVESTING PARTNER UNIT              $      426        $      697
                                                    ==========        ==========
 WEIGHTED AVERAGE INVESTING PARTNER
    UNITS OUTSTANDING                                  1,141.0           1,184.2
                                                    ==========        ==========
</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,
                                                                               -------------------------------
                                                                                  1999                 1998
                                                                               -----------         -----------

<S>                                                                            <C>                 <C>        
 CASH FLOWS FROM OPERATING ACTIVITIES:
    Net income                                                                 $   717,643         $ 1,142,365
    Adjustments to reconcile net income to net cash
       provided by operating activities:
          Depreciation, depletion and amortization                                 673,620             722,475
          Changes in operating assets and liabilities:
              Decrease in accrued revenues receivable                              289,878             114,253
              Increase (decrease) in accrued operating expenses payable              2,407             (90,825)
              Increase (decrease) in receivable from/payable to
                  Apache Corporation                                              (195,069)              1,185
                                                                               -----------         -----------
          Net cash provided by operating activities                              1,488,479           1,889,453
                                                                               -----------         -----------

 CASH FLOWS FROM INVESTING ACTIVITIES:
    Additions to oil and gas properties                                           (504,293)         (1,427,745)
    Non-cash portion of oil and gas property additions                             125,490            (196,202)
    Proceeds from sales of oil and gas properties                                       --             363,534
    Decrease in drilling advances                                                       --              13,141
                                                                               -----------         -----------
          Net cash used in investing activities                                   (378,803)         (1,247,272)
                                                                               -----------         -----------

 CASH FLOWS FROM FINANCING ACTIVITIES:
    Distributions to Investing Partners                                           (855,728)                 --
    Distributions to Managing Partner, net                                        (331,326)           (273,370)
                                                                               -----------         -----------
          Net cash used in financing activities                                 (1,187,054)           (273,370)
                                                                               -----------         -----------

 NET INCREASE (DECREASE) IN CASH AND
    CASH EQUIVALENTS                                                               (77,378)            368,811

 CASH AND CASH EQUIVALENTS, BEGINNING OF YEAR                                    1,324,949             691,797
                                                                               -----------         -----------

 CASH AND CASH EQUIVALENTS, END OF PERIOD                                      $ 1,247,571         $ 1,060,608
                                                                               ===========         ===========
 </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,
                                                                           1999                  1998
                                                                       -------------         -------------

<S>                                                                    <C>                   <C>          
                                  ASSETS

 CURRENT ASSETS:
    Cash and cash equivalents                                          $   1,247,571         $   1,324,949
    Accrued revenues receivable                                            1,008,121             1,297,999
    Receivable from Apache Corporation                                       125,096                    --
                                                                       -------------         -------------
                                                                           2,380,788             2,622,948
                                                                       -------------         -------------

 OIL AND GAS PROPERTIES, on the basis of full cost accounting:
    Proved properties                                                    168,835,993           168,331,700
    Less - Accumulated depreciation, depletion and amortization         (161,787,714)         (161,114,094)
                                                                       -------------         -------------
                                                                           7,048,279             7,217,606
                                                                       -------------         -------------
                                                                       $   9,429,067         $   9,840,554
                                                                       =============         =============

                     LIABILITIES AND PARTNERS' CAPITAL

 CURRENT LIABILITIES:
    Accrued exploration and development                                $     717,410         $     591,920
    Accrued operating expenses payable and other                             100,774                98,367
    Payable to Apache Corporation                                                 --                69,973
                                                                       -------------         -------------
                                                                             818,184               760,260
                                                                       -------------         -------------

 PARTNERS' CAPITAL:
    Managing Partner                                                         497,495               597,060
    Investing Partners (1,141.0 and 1,141.0 units
       outstanding, respectively)                                          8,113,388             8,483,234
                                                                       -------------         -------------
                                                                           8,610,883             9,080,294
                                                                       -------------         -------------
                                                                       $   9,429,067         $   9,840,554
                                                                       =============         =============
</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.   RECEIVABLE FROM/PAYABLE TO APACHE

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


2.   RIGHT OF PRESENTMENT

     As provided in the Partnership Agreement, as amended (the Amended
Partnership Agreement), a first right of presentment offer for 1999 is expected
to be made to Investing Partners in May, based on a valuation date of December
31, 1998. The Partnership is not in a position to predict how many Units will
be presented for repurchase under the May 1999 offer and cannot, at this time,
determine if the Partnership will have sufficient funds available to repurchase
all Units presented. 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.


3.   SUBSEQUENT EVENT

     On April 29, 1999, the Managing Partner entered into an agreement with
Shell Offshore Inc. and affiliated Shell entities (Shell) to purchase Shell's
interest in certain Gulf of Mexico properties, together with certain
production-related assets and proprietary 3D seismic data. The purchase price,
subject to adjustment for production since March 1, 1999, is $715 million in
cash and one million shares of the common stock of Apache. The Shell
transaction is expected to be completed in May 1999. The impact of this
acquisition on the Partnership is not known at this time.

                                       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 $.7 million in the first quarter of
1999, versus $1.1 million in the prior year period. Net income per Investing
Partner Unit decreased 39 percent, from $697 per Unit to $426 per Unit. Lower
natural gas and crude oil prices contributed to the decline.

     Revenues decreased 24 percent, from $2.2 million in the first quarter of
1998 to $1.7 million in the first quarter of 1999. Natural gas and crude oil
sales represented 80 percent and 19 percent, respectively, of the Partnership's
total revenues.

     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,                      
                                                     ----------------------------                INCREASE
                                                      1999                 1998                 (DECREASE)
                                                     -------              -------               ----------

<S>                                                    <C>                  <C>                      <C>
           Gas volume - Mcf per day                    9,241                9,079                    2%
           Average gas price - per Mcf                 $1.62                $2.19                  (26%)
           Oil volume - barrels per day                  317                  333                   (5%)
           Average oil price - per barrel            $ 11.17              $ 13.50                  (17%)
</TABLE>


FIRST QUARTER 1999 COMPARED TO FIRST QUARTER 1998

     Natural gas production revenues for the first quarter of 1999 totaled $1.4
million, 25 percent lower than the first quarter of 1998. Natural gas prices
decreased 26 percent for the first quarter of 1999 compared to the year-earlier
period, which negatively impacted revenue by $.5 million.

     The Partnership's crude oil production revenues for the first quarter of
1999 totaled $.3 million, a 21 percent decrease from the first quarter of 1998.
A 17 percent decrease in the average realized oil price led to the decline.


OPERATING EXPENSES

     The Partnership's depreciation, depletion and amortization (DD&A) rate,
expressed as a percentage of oil and gas production revenues, was approximately
40 percent during the first quarter of 1999, increasing from 33 percent during
the same period in 1998. The increase in the DD&A rate was a result of
generally declining natural gas and crude oil prices.

     Lease operating expense (LOE) was 25 percent lower in the first quarter
1999 compared to the first quarter 1998. This variance was the result of lower
workover activity in the first three months of 1999.

                                       5

<PAGE>   7


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 $1.5 million for the first three months of
1999, a decrease of 21 percent from a year ago, caused by lower oil and gas
prices. Future cash flows will be influenced similarly by fluctuations in
product prices, production levels and operating costs.

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 1999, the Partnership's oil and gas
property additions totaled $.5 million. These additions related primarily to
recompletions at South Timbalier Block 295 and drilling activity at North Padre
Block 969. Based on information supplied by the operators of the properties,
the Partnership anticipates capital expenditures of approximately $1.4 million
for the remainder of 1999. The anticipated capital expenditures relate to
completions at North Padre Block 969 and recompletions at South Timbalier Block
295. 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 $750 per Unit on March 18, 1999.
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 Partnership Agreement, as amended (the Amended
Partnership Agreement), a first right of presentment offer for 1999 is expected
to be made to Investing Partners in May, based on a valuation date of December
31, 1998. The Partnership is not in a position to predict how many Units will
be presented for repurchase under the May 1999 offer and cannot, at this time,
determine if the Partnership will have sufficient funds available to repurchase
all Units presented. 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.

     On April 29, 1999, the Managing Partner entered into an agreement with
Shell Offshore Inc. and affiliated Shell entities (Shell) to purchase Shell's
interest in certain Gulf of Mexico properties, together with certain
production-related assets and proprietary 3D seismic data. The purchase price,
subject to adjustment for production since March 1, 1999, is $715 million in
cash and one million shares of the common stock of Apache. The Shell
transaction is expected to be completed in May 1999. The impact of this
acquisition on the Partnership is not known at this time.

                                       6

<PAGE>   8


IMPACT OF THE YEAR 2000 ISSUE

     The inability of some computer programs and embedded computer chips to
distinguish between the year 1900 and the year 2000 (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
as of March 1999 are as follows:

<TABLE>
<CAPTION>
                        PHASE                                COMPLETION DATE             PERCENT COMPLETE
          -----------------------------------                ---------------             ----------------

<S>                                                           <C>                              <C> 
          Organization                                             July 1998                   100%

          Assessment                                           November 1998                   100%
              Desktop Computers
              Network Hardware
              Software
              Embedded Systems
              External Trading Partners
              Building/Infrastructure Systems
              Telecommunications Systems

          Implementation/Replacement                          September 1999                    75%
              Computer Hardware
              Core Business Software
              Desktop Software
              Embedded Systems
              Building Systems

          Contact External Trading Partners                       March 1999                   100%

          Contingency Planning                                      May 1999                    90%
</TABLE>


     To date, the Managing Partner is not aware of any significant Year 2000
issues that would cause problems in the area of safety, environmental or
business interruption. Apache will assess the risks 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 that 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 September 1999. All critical external
trading partners have been contacted to determine Year 2000 readiness and
contingency plans will be developed where assurance of Year 2000 compliance was
not received by March 31, 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 implementation of the business software project was completed and
operational effective with January 1999 production.

                                       7

<PAGE>   9


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

     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 Apache 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.

                                       8

<PAGE>   10


                          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.

                                       9

<PAGE>   11


                                   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, 1999                   /s/ Roger B. Plank
                                      -----------------------------------------
                                      Roger B. Plank
                                      Vice President and Chief Financial Officer


Dated: May 12, 1999                   /s/ Thomas L. Mitchell
                                      ------------------------------------------
                                      Thomas L. Mitchell
                                      Vice President and Controller
                                      (Chief Accounting Officer)



<PAGE>   12


                               INDEX TO EXHIBITS



<TABLE>
<CAPTION>
EXHIBIT
NUMBER                                 DESCRIPTION
------                                 -----------

<S>                                    <C>
 27.1                                  Financial Data Schedule
</TABLE>