Spot kicks were required after the two sides were all square at 2-2 after normal time in Montpellier. The holders will learn on Thursday the identity of their opponents at the Stade de France next week, with fourth-tier Rumilly-Vallieres facing Monaco in the second last four tie.
Mbappe had PSG back in front shortly after the restart. But just as it looked as though PSG were home and hosed Algeria forward Andy Delort nabbed the equaliser with seven minutes left on the clock to set up the penalty conclusion.
At 5-5 Junior Sambia sent his spot-kick high leaving Moise Kean to send PSG to the final. The French Cup has taken on added importance to Mauricio Pochettino's team as it could now be their best remaining chance to end the season with a trophy.
Their Ligue 1 domination is under severe threat as they lag three points behind leaders Lille with two games left, following their elimination from the Champions League in the semi-finals.
Standout matches today