Exhibit 5.1

October 16, 2020

Origin Bancorp, Inc.

500 South Service Road East

Ruston, Louisiana 71270

Ladies and Gentlemen:

We have acted as counsel to Origin Bancorp, Inc., a Louisiana corporation (the “Company”), in connection with the issuance and sale by the Company under the Securities Act of 1933, as amended (the “Securities Act”), of $80,000,000 in aggregate principal amount of the Company’s 4.50% Fixed-to-Floating Rate Subordinated Notes due 2030 (the “Notes”) pursuant to (i) a Registration Statement on Form S-3, Registration No. 333-234068, which was filed with the United States Securities and Exchange Commission (the “Commission”) on October 3, 2019 and became effective on October 3, 2019 (the “Registration Statement”), including the related prospectus therein, (ii) a preliminary prospectus supplement dated October 7, 2020, filed with the Commission pursuant to Rule 424(b) promulgated under the Securities Act, and (iii) a prospectus supplement dated October 8, 2020, filed with the Commission pursuant to Rule 424(b). The Notes are being sold pursuant to the underwriting agreement, dated October 8, 2020, between Stephens Inc., as underwriter, and the Company (the “Underwriting Agreement”), and issued pursuant to the Subordinated Indenture, dated as of October 16, 2020 (the “Base Indenture”), between the Company and U.S. Bank National Association, as trustee (the “Trustee”), as supplemented by the First Supplemental Indenture, dated as of October 16, 2020, between the Company and the Trustee (the “First Supplemental Indenture” and, together with the Base Indenture, the “Indenture”).

We have reviewed such 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 documents submitted to us conform to the originals. We have assumed further that the Trustee has duly authorized, executed and delivered the Indenture.

We have assumed further that (i) the Company is a corporation duly organized, validly existing and in good standing under the laws of the State of Louisiana, (ii) the Company has all requisite power, authority and legal right to execute, deliver and perform its obligations under the Indenture and the Notes, (iii) the Company has duly authorized the Indenture and the Notes and (iv) insofar as the laws of the State of Louisiana are concerned, the Company has duly executed and delivered the Indenture and the Notes.

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.

Based upon the foregoing, and subject to the qualifications set forth below, we are of the opinion that when the Notes have been duly executed by the Company and duly authenticated and delivered by the Trustee in accordance with the terms of the Indenture and issued and delivered by the Company against payment therefor as specified in the Underwriting Agreement, the Notes will constitute the valid and binding obligations of the Company, enforceable against the Company in accordance with their terms, subject to bankruptcy, insolvency, fraudulent transfer, reorganization, moratorium and other laws of general applicability relating to or affecting creditors’ rights and to general equity principles.

We are members of the bar of the District of Columbia and 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.

We hereby consent to the filing of this opinion as Exhibit 5.1 to the Company’s Current Report on Form 8-K filed on the date hereof and incorporated by reference into the Registration Statement. We also hereby consent to the reference to our firm under the heading “Legal Matters” in the prospectus constituting part of the Registration Statement. 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 or the rules and regulations of the Commission.

Very truly yours,

/s/ Covington & Burling LLP