Exhibit 5.1

600 Travis, Suite 4200 Houston, Texas 77002 +1.713.220.4200 Phone +1.713.220.4285 Fax andrewskurth.com

January 5, 2017

Cheniere Corpus Christi Holdings, LLC

700 Milam Street, Suite 1900

Houston, Texas 77002

Re: Cheniere Corpus Christi Holdings, LLC Registration Statement on Form S-4

Ladies and Gentlemen:

We have acted as special counsel to Cheniere Corpus Christi Holdings, LLC, a Delaware limited liability company (the “Company”), in connection with the public offering (the “Exchange Offer”) of (i) $1,250,000,000 aggregate principal amount of the Company’s 7.000% Senior Secured Notes due 2024 (the “New 2024 Notes”) to be issued in exchange for a like principal amount of issued and outstanding 7.000% Senior Secured Notes due 2024 (the “Old 2024 Notes”), as contemplated by the Registration Rights Agreement dated as of May 18, 2016, by and between the Company and the Initial Purchasers party thereto (the “2024 Registration Rights Agreement”) and (ii) $1,500,000,000 aggregate principal amount of issued and outstanding 5.875% Senior Secured Notes due 2025 (the “New 2025 Notes” and, together with the New 2024 Notes, the “New Notes”) to be issued in exchange for a like principal amount of issued and outstanding 5.875% Senior Secured Notes due 2025 (the “Old 2025 Notes” and, together with the Old 2024 Notes, the “Old Notes”), as contemplated by the Registration Rights Agreement dated as of December 9, 2016 by and between the Company and the Initial Purchasers Party thereto (the “2025 Registration Rights Agreement” and, together with the 2024 Registration Rights Agreement, the “Registration Rights Agreements”); in each case guaranteed (the “Guarantees”) by each of Corpus Christi Liquefaction, LLC, Cheniere Corpus Christi Pipeline, L.P. and Corpus Christi Pipeline GP, LLC (collectively, the “Guarantors” and, together with the Company, the “Registrants”). The New Notes and the Guarantees are referred to collectively herein as the “Securities.” The Securities are to be issued under the Indenture dated as of May 18, 2016, by and between the Company and The Bank of New York Mellon, as trustee (the “Trustee”), as supplemented by the First Supplemental Indenture, dated as of December 9, 2016, (as so supplemented, the “Indenture”).

This opinion is being furnished in accordance with the requirements of Item 601(b)(5) of Regulation S-K under the Securities Act.

In connection with this opinion, we have examined originals or copies, certified or otherwise identified to our satisfaction, of:

ANDREWS KURTH KENYON LLP

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Cheniere Corpus Christi Holdings, LLC

January 5, 2017

Page 2

(a) the registration statement on Form S-4 of the Company, filed with the Securities and Exchange Commission (the “SEC”) on January 5, 2017 (the “Registration Statement”);

(b) the Registration Rights Agreements;

(c) the Indenture; and

(d) the forms of the New Notes attached to the Indenture.

We have also examined originals or copies, certified or otherwise identified to our satisfaction, of such records of the Registrants and such agreements, certificates of public officials, certificates of officers or other representatives of the Registrants and others, and such other documents, certificates and records as we have deemed necessary or appropriate as a basis for the opinions set forth herein. In rendering the opinion set forth below, we have assumed and have not verified (i) the genuineness of the signatures on all documents that we have examined, (ii) the legal capacity of all natural persons, (iii) the authenticity of all documents supplied to us as originals, (iv) the conformity to the authentic originals of all documents supplied to us as certified, photostatic or faxed copies, (v) the authenticity of the originals of such latter documents and (vi) that the Registration Statement, and any amendments thereto (including post-effective amendments), will have become effective and the Securities will be issued in compliance with applicable federal and state securities laws and in the manner described in the Registration Statement. In conducting our examination of documents, we have assumed the power, corporate or other, of all parties thereto other than the Registrants to enter into and perform all obligations thereunder and have also assumed the due authorization by all requisite action, corporate or other, and the due execution and delivery by such parties of such documents and that, except as set forth in the opining paragraphs below, to the extent such documents purport to constitute agreements, such documents constitute valid and binding obligations of such parties.

We express no opinion other than as to (i) the laws of the State of New York that are normally applicable to transactions of the type contemplated by the Exchange Offer and the Securities, (ii) the Delaware Limited Liability Company Act and (iii) the Delaware Revised Uniform Limited Partnership Act (including in the case of the immediately preceding clauses (ii) and (iii), the statutory provisions, all applicable provisions of the Delaware Constitution and reported judicial decisions interpreting such laws or acts).

Based upon and subject to the foregoing and the limitations, qualifications, exceptions and assumptions set forth herein, we are of the opinion that when the New Notes (in the form examined by us) have been duly executed by the Registrants, authenticated by the Trustee in accordance with the terms of the Indenture and issued and delivered upon consummation of the Exchange Offer against receipt of Old Notes surrendered in exchange therefor in accordance with the terms of the Exchange Offer, the Registration Rights Agreement and the Indenture, (i) the New Notes will constitute valid and legally binding obligations of the Company and (ii) the Guarantees will constitute valid and legally binding obligations of the Guarantors.

Cheniere Corpus Christi Holdings, LLC

January 5, 2017

Page 3

Our opinion expressed in clause (ii) of the immediately preceding paragraph assumes that Section 5-501.6 of the New York General Obligations Law will apply to the Guarantees. Furthermore, our opinions expressed above is subject to applicable bankruptcy, insolvency (including, without limitation, all laws relating to fraudulent transfer or conveyance), reorganization, moratorium and other similar laws affecting creditors’ rights generally and to general principles of equity (regardless of whether enforcement is sought in a proceeding in equity or at law), including, without limitation, (a) the possible unavailability of specific performance, injunctive relief or any other equitable remedy and (b) concepts of materiality, reasonableness, good faith and fair dealing, and we express no opinion herein with respect to provisions relating to severability or separability. Furthermore, (a) we express no opinion regarding the validity or effect of any provision purporting to establish any obligation of any party as absolute or unconditional regardless of the occurrence or non-occurrence or existence or non-existence of any event or other state of facts or purporting to limit the use of the Indenture in interpreting any other indenture, loan or debt agreement or vice versa, and (b) certain of the waivers included in the Indenture relating to the guaranties by the guarantors that may become party thereto may be unenforceable in whole or in part.

We hereby consent to the filing of this opinion with the SEC as an exhibit to the Registration Statement. We also consent to the reference to our firm under the caption “Legal Matters” in the Registration Statement. In giving this consent, we do not thereby admit that we are included in the category of persons whose consent is required under Section 7 of the Securities Act or the rules and regulations of the SEC. This opinion is expressed as of the date hereof, and we disclaim any undertaking to advise you of any subsequent changes in the facts stated or assumed herein or of any subsequent changes in applicable law.

Very truly yours,

/s/ Andrews Kurth Kenyon LLP