Chevron (CVX) 10-Q 2013 Q1 Quarterly report

Exhibit 32.1

CERTIFICATION PURSUANT TO SECTION 906

OF THE SARBANES-OXLEY ACT OF 2002 (18 U.S.C. SECTION 1350)

In connection with the Quarterly Report of Chevron Corporation (the “Company”) on Form 10-Q for the period ended March 31, 2013, as filed with the Securities and Exchange Commission on the date hereof (the “Report”), I, John S. Watson, Chairman and Chief Executive Officer of the Company, 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:

(1) the Report fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934; and

(2) the information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company.

/S/    JOHN S. WATSON

John S. Watson

Chairman of the Board and

Chief Executive Officer

Dated: May 2, 2013

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