Eaton Vance Floating Rate Fund (EIBLX) N-CSRS Certified semi-annual shareholder report (management investment company)

FormN-CSR Item 13(b) Exhibit

CERTIFICATION PURSUANT TO

18 U.S.C. SECTION 1350,

AS ADOPTED PURSUANT TO

SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

The undersigned hereby certify in their capacity as Treasurer and President, respectively, of Eaton Vance Mutual Funds Trust (the “Trust”) that:

(a) the Semi-Annual Report of theTruston FormN-CSR for the period ended March 31, 2020 (the “Report”) fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934, as amended; and

(b) the information contained in the Report fairly presents, in all material respects, the financial condition and the results of operations of theTrustfor such period.

A signed original of this written statement required by section 906 has been provided to the Trust and will be retained by the Trust and furnished to the Securities and Exchange Commission or its staff upon request.

Eaton Vance Mutual Funds Trust

Date: May 26, 2020

/s/ James F. Kirchner

James F. Kirchner

Treasurer

Date: May 26, 2020

/s/ Payson F. Swaffield

Payson F. Swaffield

President