Neither side was able to find the breakthrough after 120 minutes of action at the Stade Saint-Symphorien, meaning the tie was decided by spot-kicks.
And Hartock made the crucial save from Jeremy Choplin, before Abel Khaled held his nerve to slot home the winning penalty and secure Brest a last-eight tie with Auxerre.
It was the the visitors who engineered the best opening of a first half that they arguably edged, as Gaetan Courtet raced through one-on-one only to blaze over the crossbar.
Metz showed some improvement after the break with Ferjani Sassi trying his luck from distance, while at the other end Youssef Adnane came agonisingly close to the opener with a bending shot that went just wide of the post.
Extra time failed to produce a winner as the match went to penalties, and the sides were locked at 3-3 after four attempts as Metz captain Sylvain Marchal and Brest's Wilfried Moimbe both hit the woodwork.
Choplin never looked confident as he stepped up for Ligue 1's bottom club and Hartock made a fine stop, before the visitors' celebrations could begin when Choplin tucked home the winning spot-kick.
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