Prosperity Bancshares (PB) 10-K 2012 FY Annual report

Exhibit 32.1

Certification Pursuant to 18 U.S.C. Section 1350, as adopted pursuant to

Section 906 of the Sarbanes-Oxley Act of 2002

In connection with this Annual Report of Prosperity Bancshares, Inc. (the “Company”) on Form 10-K for the year ending December 31, 2012 as filed with the Securities and Exchange Commission on the date hereof (the “Report”), I, David Zalman, Chairman of the Board 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:

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 operating results of the Company.

/s/    DAVID ZALMAN        

David Zalman

Chairman of the Board and Chief Executive Officer

February 28, 2013