The Ligue 2 outfit were reduced to 10 men on 34 minutes and held on for a 0-0 draw at the Stade Jean Laville before winning the shootout 4-2.
Tunisian substitute Khaled Ayari scored the decisive goal in the shootout, ending fourth-tier Moulins' cup run.
Guinea striker Mohamed Yattara's red card just after the half-hour made life difficult for Angers away from home, but Stephane Moulin's men held on.
Yohann Eudeline and Joffrey Lobo failed to convert the first two spot-kicks of the shootout, but Angers scored their next four.
Gregory Rouchon was unable to score with Moulins' second shot as Angers – who defeated Sochaux in the round of 32 – progressed to the last four.
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