Exhibit 5.1

May 8, 2013

Texas Instruments Incorporated

12500 TI Boulevard

Dallas, Texas 75266-0199

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 $500,000,000 aggregate principal amount of 1.000% Notes due 2018 (the “2018 Notes”) and $500,000,000 aggregate principal amount of 2.250% Notes due 2023 (the “2023 Notes” and together with the 2018 Notes, the “Notes”) in an underwritten public offering pursuant to an underwriting agreement dated May 1, 2013 (the “Underwriting Agreement”) among the Company and J.P. Morgan Securities LLC and Morgan Stanley & Co. LLC (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 Officer’s Certificate to be issued pursuant thereto on or about May 8, 2013. The Company has filed with the Securities and Exchange Commission a Registration Statement on Form S-3 (File No. 333-186803, the “Registration Statement”) for the purpose of registering under the Securities Act of 1933, as amended.

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 opinions 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 upon 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 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.

In connection with the opinions 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 are each 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 the Company of the Notes (a) require no action by or in respect of, or filing with, any governmental body, agency or official and (b) 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 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.

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