Exhibit 32.1

APACHE OFFSHORE INVESTMENT PARTNERSHIP

by Apache Corporation, Managing Partner

Certification of Principal Executive Officer

and Principal Financial Officer

I, G. Steven Farris, certify pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that, to my knowledge, the quarterly report on Form 10-Q of Apache Offshore Investment Partnership for the quarterly period ending March 31, 2014, fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934 (15 U.S.C. §78m or §78o (d)) and that information contained in such report fairly represents, in all material respects, the financial condition and results of operations of Apache Offshore Investment Partnership.

I, Alfonso Leon, certify pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that, to my knowledge, the quarterly report on Form 10-Q of Apache Offshore Investment Partnership for the quarterly period ending March 31, 2014, fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934 (15 U.S.C. §78m or §78o (d)) and that information contained in such report fairly represents, in all material respects, the financial condition and results of operations of Apache Offshore Investment Partnership.