LIMITED POWER OF ATTORNEY

        Know all by these present, that the undersigned hereby constitutes and
appoints each of Stephen Gunstream and Damon Terrill (each, an "Attorney-in-
fact") of Alpha Teknova, Inc. (the "Company"), or any of them acting singly and
with full power of substitution, as the undersigned's true and lawful attorney-
in-fact to:

        1.   submit to the U.S. Securities and Exchange Commission (the "SEC") a
Form ID Application Acknowledgement on the Electronic Data Gathering, Analysis,
and Retrieval system ("EDGAR") of the SEC, including any amendments thereto, and
any other documents necessary or appropriate to obtain EDGAR codes and passwords
enabling the undersigned to make electronic filings with the SEC of reports
required by Sections 13(d), 13(g) or Section 16(a) of the Securities Exchange
Act of 1934, as amended (the "Exchange Act"), or any other rule or regulation of
the SEC.

        2.   prepare, submit, execute for, and on behalf of the undersigned, in
the undersigned's capacity as an officer, director and/or holder of 10% or more
of a registered class of securities, as applicable, of the Company (a) Schedules
13D and 13G (and any amendments thereto) in accordance with Sections 13(d) and
13(g) of the Exchange Act and the rules promulgated thereunder, (b) Forms 3, 4,
and 5 (and any amendments thereto) in accordance with Section 16(a) of the
Exchange Act and the rules promulgated thereunder, and (c) any other forms or
reports the undersigned may be required to file in connection with the
undersigned's ownership, acquisition, or disposition of securities of the
Company;

        3.   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
Schedule 13D, Schedule 13G, Form 3, 4, or 5, or other form or report (or any
amendment thereto), and timely file such schedule, form or report with the SEC
and any stock exchange or similar authority;

        4.   seek or obtain, as the undersigned's representative and on the
undersigned's behalf, information on transactions in the Company's securities
from any broker or financial institution, and the undersigned hereby authorizes
any such person to release any such information to each of the Attorneys-in-
fact and approves and ratifies any such release of information; and

        5.   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 each 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 such 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 each Attorney-in-fact, in serving in
such capacities at the request of the undersigned, is not assuming any of the
undersigned's responsibilities to comply with Section 13 and Section 16 of the
Exchange Act.

        This Power of Attorney, with respect to each of the Attorneys-in-fact,
shall remain in full force and effect until the earlier of (a) the date that the
undersigned is no longer required to file Schedules 13D and 13G and Forms 3, 4,
and 5 with respect to the undersigned's beneficial ownership and transactions in
securities issued by the Company (b) with respect to the Attorney-in-fact, this
Power of Attorney is revoked by the undersigned in a signed writing delivered to
such Attorney-in-fact, or (c) the time at which such attorney-in-fact is no
longer employed by the Company or any of its subsidiaries.

        IN WITNESS WHEREOF, the undersigned has executed this Limited Power of
Attorney as of this 22nd day of April, 2021.

                                        Signed and acknowledged:

                                        By: /s/ Matthew Lowell
                                            ------------------------------

                                        Name:   Matthew Lowell
                                               ---------------------------

                                        Title:  Chief Financial Officer
                                               ---------------------------