<PAGE>   1

                               THIRD AMENDMENT TO
                                CREDIT AGREEMENT


         THIS THIRD AMENDMENT TO CREDIT AGREEMENT (this "Amendment") dated as
of February 12, 1998, is entered into by and among THE HOUSTON EXPLORATION
COMPANY, a Delaware corporation (the "Company"); CHASE BANK OF TEXAS, N.A.
(formerly known as Texas Commerce Bank National Association), a national
banking association ("CBT"), with offices at 712 Main Street, Houston, Texas
77002, as Agent (CBT, in its capacity as Agent hereinafter called "Agent") for
itself and each of the other banks or lending institutions (collectively the
"Banks") that is a signatory hereto or which hereinafter becomes a party to the
"Credit Agreement" (hereinafter defined); and the Banks.

                            PRELIMINARY STATEMENTS:

         A.      The Company, the Agent and the Banks are parties to that
certain Credit Agreement dated as of July 2, 1996 (as amended by First
Amendment to Credit Agreement and Security Agreement dated as of August 30,
1996, and as further amended by Second Amendment to Credit Agreement dated
effective as of August 4, 1997, the "Credit Agreement") under the terms of
which the Banks agreed to make available to the Company a revolving line of
credit not to exceed, in the aggregate, $150,000,000 at any one time
outstanding.

         B.      The Company, the Banks and the Agent hereby mutually desire to
amend and modify certain terms of the Credit Agreement to, among other things
(i) reduce the amount of the Borrowing Base and the Threshold Amount effective
upon the completion of the Company's initial issuance of the "Senior
Subordinated Notes" (hereinafter defined), and (ii) permit the Company's
issuance of the Senior Subordinated Notes.

         NOW THEREFORE, in consideration of the foregoing the parties hereto
hereby agree as follows:

         Section 1.  Certain Defined Terms.  All capitalized terms used herein
(including in the preliminary statements hereof) and not otherwise defined
shall have the meanings set forth in the Credit Agreement.

         Section 2.  Amendments to Certain Definitions.  The term "Agreement"
is hereby amended to mean the Credit Agreement, as amended hereby and as the
same may from time to time be further amended or supplemented.

         Section 3.  Additional Definition.  Section 1.02 of the Credit
Agreement is hereby further amended and supplemented by adding thereto the
following new definitions, to read in their entirety as follows:

                 "Senior Subordinated Notes" shall mean those certain Senior
         Subordinated Notes of the Company to be issued by the Company on terms
         substantially similar
<PAGE>   2
         to those set forth on Annex A attached to the Third Amendment, in the
         aggregate principal amount not to exceed $150,000,000.

                 "Third Amendment" shall mean that certain Third Amendment to
         Credit Agreement dated as of February __, 1998, among the Company, the
         Agent and the Lenders party thereto.

         Section 4.  Borrowing Base; Threshold Amount.  For the period from and
after the date of completion of the Company's initial issuance of the Senior
Subordinated Notes to and including the date of the next redetermination of the
Borrowing Base in accordance with Section 2.09 of the Credit Agreement, the
amount of the Borrowing Base shall be $115,000,000.00, and the amount of the
Threshold Amount shall be $115,000,000.00.

         Section 5.  Debt.  Section 9.01 of the Credit Agreement is hereby
amended as follows:

                 (a)      by deleting the word "and" found at the end of clause
         (h) of Section 9.01;

                 (b)      by deleting the period (.) found at the end of clause
         (i) of Section 9.01 and substituting therefor "; and "; and

                 (c)      by adding thereto a new clause, to be clause (j), to
        read in its entirety as follows:

                          "(j)    Debt of the Company evidenced by the Senior
                 Subordinated Notes; provided, however, that (i) the aggregate
                 principal amount of the Senior Subordinated Notes shall not
                 exceed $150,000,000, (ii), payment of principal of, premium,
                 if any, and interest on the Senior Subordinated Notes shall be
                 subordinated to the payment of the Company's obligations under
                 this Agreement, the Notes, the Letters of Credit,  the Letter
                 of Credit Agreements and any other Security Instruments on the
                 terms provided for in Annex A attached to the Third Amendment,
                 and (iii) the portion of the proceeds received by the Company
                 from the issuance of the Senior Subordinated Notes used by the
                 Company in connection with the proposed acquisition of oil and
                 gas properties announced by the Company on February 9, 1998,
                 shall not exceed $60,000,000, and the balance of all such
                 proceeds shall be used by the Company to prepay amounts
                 outstanding under this Agreement and the Notes."

         Section 6.  Acquisitions.  The first paragraph of Section 9.11 of the
Credit Agreement is hereby amended in its entirety to read as follows:

                 "Section 9.11  Acquisitions.  Make any Acquisition other than
         an Acceptable Acquisition."

         Section 7.   Negative Pledge Agreements.  Section 9.14 of the Credit
Agreement is hereby amended by inserting a comma (,) and the phrase "the Senior
Subordinated Notes and any indenture relating thereto" immediately following
the term "Security Instruments" found in the third line thereof.




                                     -2-
<PAGE>   3
         Section 8.   Ratification.  The Company acknowledges and ratifies the
Credit Agreement as amended hereby, and agrees and acknowledges that all the
terms thereof as amended hereby (a) are hereby brought forward for the benefit
of the Banks and the Agent and (b) shall remain in full force and effect.

         Section 9.   Counterparts.  This Amendment may be signed in any number
of counterparts, each of which shall be construed as an original, but all of
which together shall constitute one and the same instrument.

         Section 10.   References.  On and after the date hereof, the terms
"Agreement," "hereof," "herein," "hereunder," and terms of like import when
used in the Credit Agreement shall, except where the context otherwise
requires, refer to the Credit Agreement, as amended and supplemented by this
Amendment.

         Section 11.   Choice of Law.  This Amendment shall be governed by and
construed in accordance with the laws of the State of Texas.

         Section 12.   Final Agreement of the Parties.  THIS AMENDMENT, THE
CREDIT AGREEMENT, THE NOTES AND OTHER SECURITY INSTRUMENTS CONSTITUTE A "LOAN
AGREEMENT" AS DEFINED IN SECTION 26.02(a) OF THE TEXAS BUSINESS AND COMMERCE
CODE, AND REPRESENT THE FINAL AGREEMENT BETWEEN THE PARTIES AND MAY NOT BE
CONTRADICTED BY EVIDENCE OF PRIOR, CONTEMPORANEOUS OR SUBSEQUENT ORAL
AGREEMENTS OF THE PARTIES.  THERE ARE NO UNWRITTEN ORAL AGREEMENTS BETWEEN THE
PARTIES.


                            [SIGNATURE PAGES FOLLOW]





                                     -3-
<PAGE>   4
         IN WITNESS WHEREOF, the parties hereto have caused this Agreement to
be executed by their respective officers thereunto duly authorized as of the
date first above written.

                                       COMPANY:
                                       
                                       THE HOUSTON EXPLORATION COMPANY
                                       
                                       
                                       By: /s/ JAMES F. WESTMORELAND
                                          -------------------------------------
                                             James F. Westmoreland
                                              Vice President





                             (Signature Page 1)
<PAGE>   5
                                       AGENT AND BANKS:

                                       CHASE BANK OF TEXAS, N.A. (formerly 
                                       known as Texas Commerce Bank National
                                       Association), Individually as a Bank and
                                       in its capacity as Agent
                                       
                                       
                                       
                                       By:  /s/ Paul J. Nidoh
                                           ------------------------------------
                                            Paul J. Nidoh
                                            Vice President
                                       
                                       
                                       CIBC INC.

                                       
                                       
                                       By:   /s/ ALELESANDRA K. DYMERAUS
                                           ------------------------------------
                                       Name:     Alelesandra K. Dymeraus
                                             ----------------------------------
                                       Title:    Authorized Signatory
                                              ---------------------------------




                             (Signature Page 2)
<PAGE>   6
                                       FIRST UNION NATIONAL BANK
                                       
                                       
                                       
                                       By: /s/ DAVID ROBERTS
                                          -------------------------------------
                                       Name:   David Roberts
                                       Title:  Senior Vice President





                             (Signature Page 3)
<PAGE>   7
                                       THE BANK OF NOVA SCOTIA
                                       
                                       By:   THE BANK OF NOVA SCOTIA,
                                             ATLANTA AGENCY
                                             
                                             
                                             
                                             By:     /s/ F.C.H. ASHBY
                                                  -----------------------------
                                             Name:   F.C.H. Ashby
                                             Title:  Senior Manager Loan 
                                                     Operations





                             (Signature Page 4)
<PAGE>   8
                                       PNC BANK, N.A.
                                       
                                       
                                       
                                       By:  /s/ JOHN WAY
                                           ----------------------------------
                                       Name:    John Way
                                       Title:   Commercial Banking Officer





                             (Signature Page 5)