Unifi (UFI) 10-Q 2013 Q3 Quarterly 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 the Unifi, Inc. (the “Company”) Quarterly Report on Form 10-Q for the period ended March 24, 2013 as filed with the Securities and Exchange Commission on the date hereof (the “Report”), I, William L. Jasper, 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, as amended; and

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

Date:      May 3, 2013 /s/   WILLIAM L. JASPER                                

  William L. Jasper

  Chairman of the Board and Chief Executive Officer

  (Principal Executive Officer)