LIMITED POWER OF ATTORNEY

        The undersigned hereby constitutes and appoints Shoaib Ghias, signing
singly, and with full power of substitution, the undersigned's true and lawful
attorney-in-fact to:

        (1)   execute for and on behalf of the undersigned, in the undersigned's
capacity as an officer and/or director of Zafgen, Inc. (the "Company"), from
time to time the following U.S. Securities and Exchange Commission ("SEC")
forms: (i) Form ID, including any attached documents, to effect the assignment
of codes to the undersigned to be used in the transmission of information to the
SEC using the EDGAR System; (ii) Form 3, Initial Statement of Beneficial
Ownership of Securities, including any attached documents;  (iii) Form 4,
Statement of Changes in Beneficial Ownership of Securities, including any
attached documents; (iv) Form 5, Annual Statement of Beneficial Ownership of
Securities in accordance with Section 16(a) of the Securities Exchange Act of
1934, as amended, and the rules thereunder, including any attached documents;
(v) Schedule 13D and (vi) amendments of each thereof, in accordance with the
Securities Exchange Act of 1934, as amended, and the rules thereunder, including
any attached documents;

        (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
Form 3, 4 or 5, Schedule 13D or any amendment(s) thereto, and timely file such
form(s) with the SEC and any securities exchange, national association or
similar authority; and

        (3)   take any other action of any type whatsoever in connection with
the foregoing which, in the opinion of such attorney-in-fact, may be of benefit
to, in the best interest 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 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 attorney-in-fact, in
serving in such capacity at the request of the undersigned, is not assuming, nor
is the Company assuming, any of the undersigned's responsibilities to comply
with Section 16 or Regulation 13D-G of the Securities Exchange Act of 1934, as
amended.  The undersigned hereby agrees to indemnify the attorney in fact and
the Company from and against any demand, damage, loss, cost or expense arising
from any false or misleading information provided by the undersigned to the
attorney-in fact.

        This Power of Attorney shall remain in full force and effect until the
undersigned is no longer required to file such forms with respect to the
undersigned's holdings 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 as of June 4, 2018.

                                        /s/ Robert J. Perez
                                        -------------------------------
                                        Robert J. Perez