EX-5.1

Exhibit 5.1

Covington & Burling LLP

One CityCenter

850 Tenth Street, NW

Washington, DC 20001-4956

T +1 202 662 6000

March 9, 2020

Evans Bancorp, Inc.

One Grimsby Drive

Hamburg, New York 14075

Ladies and Gentlemen:

We have acted as counsel to Evans Bancorp, Inc., a New York corporation (the “Company”), in connection with the Company’s Registration Statement on Form S-4 (File No. 333-236425) (as amended to the date hereof, the “Registration Statement”) filed pursuant to the Securities Act of 1933, as amended (the “Securities Act”). The Registration Statement registers an aggregate of 432,080 shares of the Company’s common stock, par value $0.50 per share (the “Shares”), to be issued pursuant to an Agreement and Plan of Reorganization, dated as of December 19, 2019 (as amended through the date hereof, the “Merger Agreement”), by and among the Company, MMS Merger Sub, Inc., a Maryland corporation (“Merger Sub”), and FSB Bancorp, Inc., a Maryland corporation (“FSB”), pursuant to which, among other things, (a) Merger Sub will merge with and into FSB, with FSB continuing as the surviving corporation and (b) immediately thereafter, FSB will merge with and into the Company, with the Company as the surviving corporation.

We have reviewed the Merger Agreement, the Certificate of Incorporation of the Company, as amended, the Amended and Restated By-laws of the Company, and such other corporate records, certificates and other documents, and such questions of law, as we have considered necessary or appropriate for the purposes of this opinion. We have assumed that all signatures are genuine, that all documents submitted to us as originals are authentic and that all copies of such documents submitted to us conform to the originals. We have relied as to certain matters on information obtained from public officials, officers of the Company and other sources believed by us to be responsible.

We have assumed that, prior to the issuance of any of the Shares, (a) the Registration Statement will have become effective under the Securities Act, (b) to the extent required under the laws of Maryland, the stockholders of FSB will have approved the Merger Agreement, including the transactions contemplated thereby, and (c) the transactions contemplated by the Merger Agreement will have been consummated in accordance with the Merger Agreement and in conformity with applicable requirements under Maryland law.

Based upon the foregoing, we are of the opinion that the Shares have been duly authorized and, when the Registration Statement has become effective under the Securities Act, the Shares, when duly issued in accordance with the Merger Agreement, will be validly issued, fully paid and non-assessable.

We are members of the bar of the State of New York. We do not express any opinion herein on any laws other than the law of the State of New York.