EXHIBIT 5.1

2828 N. Harwood Street

Suite 1800

Dallas, TX 75201-2139

214-922-3400

Fax: 214-922-3899

www.alston.com

December 2, 2016

Texas Capital Bancshares, Inc.

2000 McKinney Avenue, Suite 700

Dallas, Texas 75201

Ladies and Gentlemen:

We have acted as counsel to Texas Capital Bancshares, Inc., a Delaware corporation (the “Company”), and have advised the Company in connection with the offer and sale by the Company of 3,450,000 shares of the Company’s common stock, par value $0.01 per share (the “Shares”), pursuant to the Company’s Registration Statement on Form S-3 (Registration No. 333-196339), (the “Registration Statement”), filed by the Company with the Securities and Exchange Commission (the “Commission”) under the Securities Act of 1933, as amended (the “Securities Act”). A prospectus supplement dated November 28, 2016, which together with the prospectus filed with the Registration Statement shall constitute the “Prospectus,” has been filed pursuant to Rule 424(b) promulgated under the Securities Act.

In connection with rendering this opinion, we have examined originals or copies of (i) the Registration Statement, (ii) the Prospectus, (iii) the Certificate of Incorporation of the Company, and the Amended and Restated Bylaws of the Company, each as amended to the date hereof, (iv) certain resolutions of the Board of Directors of the Company and of committees of the Board of Directors of the Company relating to the offer and sale of the Shares and related matters, (v) the form of specimen certificate representing the Company’s common stock, and (vi) such other documents and records as we have deemed necessary and relevant for purposes hereof. In addition, we have relied on certificates of public officials as to certain matters of fact relating to this opinion and have made such investigations of law as we have deemed necessary and relevant as a basis hereof. In such examination, we have assumed the legal capacity of all natural persons, the genuineness of all signatures, the authenticity of all documents, certificates and records submitted to us as originals, the conformity to original documents, certificates and records of all documents, certificates and records submitted to us as copies and the truthfulness of all statements of fact contained therein. In making our examination of executed documents or documents to be executed, we have assumed that they constitute or will constitute valid, binding and enforceable obligations of such parties, other than the Company. We have also assumed that all Shares will be issued and sold in the manner described in the Prospectus and in accordance with the terms of the Underwriting Agreement dated November 28, 2016 relating to the offer and sale of the Shares (the “Underwriting Agreement”).

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Texas Capital Bancshares, Inc.

December 2, 2016

Page 2

Based on the foregoing, and subject to the limitations, assumptions and qualifications set forth herein, and having due regard for such legal considerations as we deem relevant, we are of the opinion that (i) the issuance of the Shares by the Company in accordance with the terms of the Underwriting Agreement has been duly authorized and (ii) when the Shares have been duly issued and delivered by the Company in accordance with the terms of the Underwriting Agreement, the Shares will be validly issued, fully paid and non-assessable.

The foregoing opinion is based on and is limited to the General Corporation Law of the State of Delaware and the relevant law of the United States of America, and we render no opinion with respect to the law of any other jurisdiction. The reference and limitation to “General Corporation Law of the State of Delaware” includes the statutory provisions and all applicable provisions of the Delaware Constitution and reported judicial decisions interpreting these laws. We are not admitted to the practice of law in the State of Delaware.

We hereby consent to the filing of this opinion as an exhibit to a Current Report on Form 8-K and to the reference to us under the heading “Legal Matters” in the Prospectus. 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 Act.

Very Truly Yours,

/s/ Alston & Bird LLP

ALSTON & BIRD LLP