Exhibit 5.1

May 6, 2021

Texas Capital Bancshares, Inc.,

2000 McKinney Avenue, Suite 700,

Dallas, Texas 75201.

Ladies and Gentlemen:

In connection with the registration under the Securities Act of 1933 (the “Act”) of $375,000,000 aggregate principal amount of 4.000% Fixed-to-Fixed Rate Subordinated Notes due 2031 (the “Securities”) of Texas Capital Bancshares, Inc., a Delaware corporation (the “Company”), we, as your counsel, have examined such corporate records, certificates and other documents, and such questions of law, as we have considered necessary or appropriate for purposes of this opinion.

Upon the basis of such examination, it is our opinion that the Securities constitute valid and legally binding obligations of the Company, subject to bankruptcy, insolvency, fraudulent transfer, reorganization, moratorium and similar laws of general applicability relating to or affecting creditors’ rights and to general equity principles.

In rendering the foregoing opinion, we are not passing upon, and assume no responsibility for, any disclosure in any registration statement or any related prospectus or other offering material relating to the offer and sale of Securities.

The foregoing opinion is limited to the Federal laws of the United States, the laws of the State of New York and the General Corporation Law of the State of Delaware, and we are expressing no opinion as to the effect of the laws of any other jurisdiction.

We have relied as to certain factual matters on information obtained from public officials, officers of the Company and other sources believed by us to be responsible, and we have assumed that the Indenture, dated September 21 2012, as supplemented by the First Supplemental Indenture, dated May 6, 2021 (the “First Supplemental Indenture” and, together with the Base Indenture, the “Indenture”) has been duly authorized, executed and delivered by U.S. Bank, National Association hereunder, assumptions which we have not independently verified.

We hereby consent to the filing of this opinion as an exhibit to a Current Report on Form 8-K to be incorporated by reference into the Registration Statement relating to the Securities and to references to us under the heading “Legal Matters” in the Prospectus Supplement relating to the Securities, dated April 29, 2021. In giving such

consent, we do not thereby admit that we are in the category of persons whose consent is required under Section 7 of the Act.