Exhibit 5.1

Concert Pharmaceuticals, Inc.

99 Hayden Avenue, Suite 500

Lexington, Massachusetts 02421

Ladies and Gentlemen:

This opinion is furnished to you in connection with (i) the Registration Statement on Form S-3 (File No. 333-202451) (the “Registration Statement”) filed by Concert Pharmaceuticals, Inc., a Delaware corporation (the “Company”), with the Securities and Exchange Commission (the “Commission”) under the Securities Act of 1933, as amended (the “Securities Act”), for the registration of, among other things, shares of the Company’s common stock, $0.001 par value per share (the “Common Stock”), which may be issued from time to time on a delayed or continuous basis pursuant to Rule 415 under the Securities Act at an aggregate initial offering price not to exceed $125,000,000, as set forth in the Registration Statement and the prospectus contained therein (the “Base Prospectus”) and any amendments or supplements thereto; (ii) the preliminary prospectus supplement, dated March 18, 2015 (the “Preliminary Prospectus Supplement”); and (iii) the prospectus supplement, dated March 19, 2015 (the “Prospectus Supplement” and, together with the Preliminary Prospectus Supplement and the Base Prospectus, the “Prospectuses”) relating to the issuance and sale pursuant to the Registration Statement of up to 3,795,000 shares of Common Stock (the “Shares”), including up to 495,000 shares of Common Stock issuable upon exercise of an option to purchase additional shares granted by the Company.

The Shares are to be issued and sold by the Company pursuant to an underwriting agreement dated March 19, 2015 (the “Underwriting Agreement”), among the Company and Wells Fargo Securities, LLC and JMP Securities LLC, as representatives of the several underwriters named therein, the form of which is being filed with the Commission as Exhibit 1.1 to the Company’s Current Report on Form 8-K, filed on the date hereof.

We are acting as counsel for the Company in connection with the issue and sale by the Company of the Shares. We have examined a signed copy of the Registration Statement and copies of the Prospectuses, each as filed with the Commission. We have also examined and relied upon the Underwriting Agreement, minutes of meetings of the stockholders and the Board of Directors of the Company, including committees thereof, as provided to us by the Company, the Certificate of Incorporation and By-laws of the Company, each as restated and/or amended to date, and such other documents as we have deemed necessary for purposes of rendering the opinions hereinafter set forth.

In our examination of the foregoing documents, we have assumed the genuineness of all signatures, the authenticity of all documents submitted to us as originals, the conformity to original documents of all documents submitted to us as copies, the authenticity of the originals of such latter documents and the legal competence of all signatories to such documents.

We express no opinion herein as to the laws of any state or jurisdiction other than the General Corporation Law of the State of Delaware and the federal laws of the United States of America.

Based upon and subject to the foregoing, we are of the opinion that the Shares have been duly authorized for issuance and, when the Shares are issued and paid for in accordance with the terms and conditions of the Underwriting Agreement, the Shares will be validly issued, fully paid and non-assessable.

Please note that we are opining only as to the matters expressly set forth herein, and no opinion should be inferred as to any other matters. This opinion is based upon currently existing statutes, rules, regulations and judicial decisions, and we disclaim any obligation to advise you of any change in any of these sources of law or subsequent legal or factual developments which might affect any matters or opinions set forth herein.

We hereby consent to the filing of this opinion with the Commission as an exhibit to the Current Report on Form 8-K to be filed by the Company in connection with the issue and sale of the Shares in accordance with the requirements of Item 601(b)(5) of Regulation S-K under the Securities Act and to the use of our name in the Prospectus Supplement under the caption “Legal Matters.” In giving such consent, we do not hereby admit that we are in the category of persons whose consent is required under Section 7 of the Securities Act or the rules and regulations of the Commission.