POWER OF ATTORNEY

        Know all by these present, that the undersigned hereby constitutes and
appoints each of Ankit Mahadevia, M.D., Satyavrat Shukla, Tamara Joseph, and
Meghan Small of Spero Therapeutics, Inc. (the "Company"), and Mathew J.
Gardella and Matthew W. Tikonoff of Mintz, Levin, Cohn, Ferris, Glovsky and
Popeo, P.C., signing singly, with full power of substitution, the undersigned's
true and lawful attorney-in-fact to:

        (1)   execute for and on behalf of the undersigned, forms and
              authentication documents for EDGAR Filing Access;

        (2)   do and perform any and all acts for and on behalf of the
              undersigned which may be necessary or desirable to complete and
              execute any such forms and authentication documents;

        (3)   execute for and on behalf of the undersigned, in the
              undersigned's capacity as an officer, director and/or 10%
              shareholder of the Company, Forms 3, 4, and 5 in accordance with
              Section 16(a) of the Securities and Exchange Act of 1934 and the
              rules thereunder;

        (4)   do and perform any and all acts for and on behalf of the
              undersigned which may be necessary or desirable to complete and
              execute any such Form 3, 4, or 5 and timely file such form with
              the United States Securities and Exchange Commission and any
              stock exchange or similar authority; and

        (5)   take any other action of any type whatsoever in connection with
              the foregoing which, in the option of such attorney-in-fact, may
              be of benefit to, in the best interests of, or legally required
              by the undersigned, it being understood that the documents
              executed by such attorney-in-fact on behalf of the undersigned
              pursuant to this Power of Attorney shall be in such form and
              shall contain such terms and conditions as such attorney-in-fact
              may approve in such attorney-in-fact's discretion.

        The undersigned hereby grants to each such attorney-in-fact full power
and authority to do and perform any and every act and thing whatsoever
requisite, necessary, or proper to be done in the exercise of any of the rights
and powers herein granted, as fully to all intents and purposes as the
undersigned might or could do if personally present, with full power of
substitution or revocation, hereby ratifying and confirming all that such
attorney-in-fact, or such attorney-in-fact's substitute or substitutes, shall
lawfully do or cause to be done by virtue of this power of attorney and the
rights and powers herein granted. The undersigned acknowledges that the
foregoing attorneys-in-fact, in serving in such capacity at the request of the
undersigned, are not assuming, nor is the Company assuming, any of the
undersigned's responsibilities to comply with Section 16 of the Securities
Exchange Act of 1934, as amended.

        This Power of Attorney shall remain in full force and effect until the
undersigned is no longer required to file Forms 3, 4, and 5 with respect to the
undersigned's holding of and transactions in securities issued by the Company,
unless earlier revoked by the undersigned in a signed writing delivered to the
foregoing attorneys-in-fact.

        IN WITNESS WHEREOF, the undersigned has caused this Power of Attorney
to be executed on this 10th day of September, 2021.

                                   /s/ Timothy Keutzer
                                   -------------------------------
                                   Timothy Keutzer