Exhibit 5.1

New York Menlo Park Washington DC São Paulo London Paris Madrid Tokyo Beijing Hong Kong

Davis Polk & Wardwell LLP 1600 El Camino Real Menlo Park, CA 94025 650 752 2000 tel 650 752 2111 fax

May 4, 2017

Texas Instruments Incorporated

12500 TI Boulevard

Dallas, Texas 75243

Ladies and Gentlemen:

We have acted as special counsel for Texas Instruments Incorporated, a Delaware corporation (the “Company”), in connection with the Company’s offering of $300,000,000 aggregate principal amount of 2.750% Notes due 2021 (the “2021 Notes”) and $300,000,000 aggregate principal amount of 2.625% Notes due 2024 (the “2024 Notes” and, together with the 2021 Notes, the “Notes”) in an underwritten public offering pursuant to an underwriting agreement dated April 27, 2017 (the “Underwriting Agreement”) among the Company and Citigroup Global Markets Inc., Morgan Stanley & Co. LLC and MUFG Securities Americas Inc. (the “Representatives”), as representatives of the several underwriters listed in Schedule II thereto (the “Underwriters”). The Notes are to be issued pursuant to an Indenture dated as of May 23, 2011 (the “Indenture”) by and between the Company and U.S. Bank National Association, as Trustee, and an Officers’ Certificate to be issued pursuant thereto on or about May 4, 2017. The Company has filed with the Securities and Exchange Commission a Registration Statement on Form S-3 (File No. 333-209678, the “Registration Statement”) for the purpose of registering under the Securities Act of 1933, as amended (“the Securities Act”), the offer and sale of certain securities, including the Notes. The 2021 Notes are to be a further issuance of and form a single series with the existing 2.750% Notes due 2021 issued by the Company on March 12, 2014.

We, as your counsel, have examined originals or copies of such documents, corporate records, certificates of public officials and other instruments as we have deemed necessary or advisable for the purpose of rendering this opinion.

In rendering the opinion expressed herein, we have, without independent inquiry or investigation, assumed that (i) all documents submitted to us as originals are authentic and complete, (ii) all documents submitted to us as copies conform to authentic, complete originals, (iii) all documents filed as exhibits to the Registration Statement that have not been executed will conform to the forms thereof, (iv) all signatures on all documents that we reviewed are genuine, (v) all natural persons executing documents had and have the legal capacity to do so, (vi) all statements in certificates of public officials and officers of the Company that we reviewed were and are accurate and (vii) all representations made by the Company as to matters of fact in the documents that we reviewed were and are accurate.

Based on the foregoing, and subject to the additional assumptions and qualifications set forth below, we advise you that, in our opinion, the Notes have been duly authorized in accordance with the Indenture, and, when executed, authenticated and issued in accordance with the provisions of the Indenture and delivered to and paid for by the Underwriters in accordance with

Texas Instruments Incorporated 2 May 4, 2017

the terms of the Underwriting Agreement, such Notes will constitute valid and binding obligations of the Company, enforceable in accordance with their terms, subject to applicable bankruptcy, insolvency and similar laws affecting creditors’ rights generally, concepts of reasonableness and equitable principles of general applicability, provided that we express no opinion as to the validity, legally binding effect or enforceability of any provision that permits holders to collect any portion of stated principal amount upon acceleration of the Notes to the extent determined to constitute unearned interest.

In connection with the opinion expressed above, we have assumed that (i) the Registration Statement became effective upon filing with the Commission and such effectiveness shall not have been terminated or rescinded and (ii) at or prior to the time of the delivery of the Notes, the Indenture and the Notes (the “Documents”) are valid, binding and enforceable agreements of each party thereto (other than as expressly covered above in respect of the Company). We have also assumed that the execution, delivery and performance by each party to each Document to which it is a party (a) are within its corporate powers, (b) do not contravene, or constitute a default under, the certificate of incorporation or bylaws or other constitutive documents of such party, (c) require no action by or in respect of, or filing with, any governmental body, agency or official and (d) do not contravene, or constitute a default under, any public policy, any provision of applicable law or regulation or any judgment, injunction, order or decree or any agreement or other instrument binding upon such party, provided that we make no such assumption to the extent that we have specifically opined as to such matters with respect to the Company.

We are members of the Bars of the States of New York and California and the foregoing opinion is limited to the laws of the State of New York and the General Corporation Law of the State of Delaware, except that we express no opinion as to any law, rule or regulation that is applicable to the Company, the Documents or such transactions solely because such law, rule or regulation is part of a regulatory regime applicable to any party to any of the Documents or any affiliates due to the specific assets or business of such party or such affiliate.

We hereby consent to the filing of this opinion as an exhibit to a report on Form 8-K to be filed by the Company on the date hereof and further consent to the reference to our name under the caption “Validity of Securities” in the base prospectus and supplement thereto, which are a part of the Registration Statement. In giving this consent, we do not admit that we are in the category of persons whose consent is required under Section 7 of the Securities Act.

Very truly yours,

/s/ Davis Polk & Wardwell LLP