Exhibit 5.1

May 8, 2013

State Street Corporation

One Lincoln Street

Boston, Massachusetts 02111

Ladies and Gentlemen:

We have acted as counsel for State Street Corporation, a Massachusetts corporation (the “Company”), in connection with the offer and sale of $500,000,000 aggregate principal amount of the Company’s 1.350% Senior Notes due 2018 (the “Senior Notes”) and $1,000,000,000 aggregate principal amount of the Company’s 3.100% Senior Subordinated Notes due 2023 (the “Subordinated Notes,” and together with the Senior Notes, the “Debt Securities”), pursuant to an Underwriting Agreement, dated as of May 8, 2013 (the “Underwriting Agreement”), among the Company and Merrill Lynch, Pierce, Fenner & Smith Incorporated, Goldman, Sachs & Co. and Morgan Stanley & Co. LLC, as representatives of the several underwriters named in the Underwriting Agreement. The Senior Notes will be issued pursuant to an Indenture (the “Senior Notes Indenture”), dated as of March 11, 2009, between the Company and U.S. Bank National Association, a national banking association organized under the laws of the United States, as trustee (the “Senior Notes Trustee”). The Subordinated Notes will be issued pursuant to an Indenture (the “Subordinated Notes Indenture,” and together with the Senior Notes Indenture, the “Indentures”), dated as of March 11, 2009, between the Company and Wells Fargo Bank, National Association, a national banking association duly organized and existing under the laws of the United States, as trustee (the “Subordinated Notes Trustee”).

As such counsel, we have assisted in the preparation and filing with the Securities and Exchange Commission (the “Commission”) of the Company’s prospectus supplement dated May 8, 2013 (the “Prospectus Supplement”) to the prospectus, dated March 2, 2012 (the “Base Prospectus”), each relating to the registration statement on Form S-3 (File No. 333-179875) (the “Registration Statement”) filed by the Company with the Commission on March 2, 2012.

We have examined and relied upon corporate or other proceedings of the Company regarding (i) the authorization of the execution and delivery of the Indentures, the Underwriting Agreement and the issuance of the Debt Securities, (ii) the Registration Statement, (iii) the Base Prospectus, (iv) the Prospectus Supplement, (v) the Underwriting Agreement and (vi) the Indentures. We have also examined and relied upon originals or copies, certified or otherwise identified to our satisfaction, of such other corporate records of the Company, such other agreements and instruments, certificates of public officials, officers of the Company and other persons, and such other documents, instruments and certificates as we have deemed necessary as a basis for the opinions hereinafter expressed. Insofar as this opinion relates to factual matters,

Wilmer Cutler Pickering Hale and Dorr LLP, 399 Park Avenue, New York, New York 10022

Beijing    Berlin    Boston    Brussels    Frankfurt     London    Los Angeles    New York    Oxford    Palo Alto    Waltham    Washington

State Street Corporation

May 8, 2013

Page 2

we have assumed with your permission and without independent investigation that the statements of the Company contained in the Registration Statement are true and correct as to all factual matters stated therein.

In our examination of the documents referred to above, we have assumed the genuineness of all signatures, the legal capacity of all individual signatories, the authenticity of all documents submitted to us as originals, the conformity to original documents of all documents submitted to us as copies, the authenticity of such original documents, and the completeness and accuracy of the corporate records of the Company provided to us by the Company.

In rendering the opinions set forth below, we have assumed that (i) the Senior Notes Trustee has the power, corporate or other, to enter into and perform its obligations under the Senior Notes Indenture, (ii) the Senior Notes Indenture will be a valid and binding obligation of the Senior Notes Trustee, and (iii) the Senior Notes Trustee shall have been qualified under the Trust Indenture Act of 1939, as amended. We have also assumed that (A) the Subordinated Notes Trustee has the power, corporate or other, to enter into and perform its obligations under the Subordinated Notes Indenture, (B) the Subordinated Notes Indenture will be a valid and binding obligation of the Subordinated Notes Trustee, and (C) the Subordinated Notes Trustee shall have been qualified under the Trust Indenture Act of 1939, as amended. We have also assumed the due authentication of the Debt Securities by the Senior Notes Trustee and Subordinated Notes Trustee, as the case may be, that there will not have occurred, prior to the date of issuance of the Debt Securities, any change in law affecting the validity or enforceability of such Debt Securities and that at the time of the issuance and sale of the Debt Securities, the Board of Directors of the Company has not taken any action to rescind or otherwise reduce its prior authorization of the issuance of the Debt Securities.

We express no opinion herein as to the laws of any jurisdiction other than the state laws of the State of New York and the federal laws of the United States of America.

We have assumed for purposes of our opinions below that no authorization, approval or other action by, and no notice to or filing with, any governmental authority or regulatory body or (to the extent the same is required under any agreement or document binding on it of which an addressee has knowledge, has received notice or has reason to know) any other third party is required for the due execution, delivery or performance by the Company of the Indentures or the Debt Securities or, if any such authorization, approval, consent, action, notice or filing is required, it will have been duly obtained, taken, given or made and will be in full force and effect.

Our opinions below are qualified to the extent that they may be subject to or affected by (i) applicable bankruptcy, insolvency, reorganization, moratorium, usury, fraudulent conveyance or similar laws relating to or affecting the rights or remedies of creditors generally, (ii) statutory or decisional law concerning recourse by creditors to security in the absence of notice or hearing, (iii) duties and standards imposed on creditors and parties to contracts, including, without

State Street Corporation

May 8, 2013

Page 3

limitation, requirements of materiality, good faith, reasonableness and fair dealing, and (iv) general equitable principles. Furthermore, we express no opinion as to the availability of any equitable or specific remedy upon any breach of the Indentures or the Debt Securities, or to the successful assertion of any equitable defenses, inasmuch as the availability of such remedies or the success of any equitable defenses may be subject to the discretion of a court. We also express no opinion herein with respect to compliance by the Company with the securities or “blue sky” laws of any state or other jurisdiction of the United States or of any foreign jurisdiction. In addition, we express no opinion and make no statement herein with respect to the antifraud laws of any jurisdiction.

On the basis of, and subject to, the foregoing, we are of the opinion that when the Debt Securities have been duly executed by the Company and duly authenticated by the Senior Notes Trustee or the Subordinated Notes Trustee, as the case may be, in accordance with the terms of the Senior Notes Indenture or the Subordinated Notes Indenture, as the case may be, and delivered to the purchasers thereof against payment of the consideration therefor duly approved by the Company, and subject to the final terms of the Debt Securities not resulting in a default under or a breach of any agreement or instrument binding upon the Company and complying with any requirement or restriction imposed by any court or governmental entity having jurisdiction over the Company, the Debt Securities will constitute valid and binding obligations of the Company, enforceable against the Company, in accordance with their terms.

Please note that we are opining only as to the matters expressly set forth herein, and no opinion should be inferred as to any other matters. This opinion is based upon currently existing statutes, rules, regulations and judicial decisions and is rendered as of the date hereof, and we disclaim any obligation to advise you of any change in any of the foregoing sources of law or subsequent developments in law or changes in facts or circumstances that might affect any matters or opinions set forth herein.

We hereby consent to the filing of this opinion with the Commission as an exhibit to the Company’s Current Report on Form 8-K to be filed on or about May 14, 2013, which Form 8-K will be incorporated by reference into the Registration Statement, and to the use of our name therein and in the related Base Prospectus and Prospectus Supplement under the caption “Legal Matters.” In giving such consent, we do not hereby admit that we are in the category of persons whose consent is required under Section 7 of the Securities Act of 1933, as amended, or the rules and regulations of the Commission.

Very truly yours,