Mangani scored before the break and Toko Ekambi wrapped it up in stoppage time after Guingamp's Jimmy Briand missed a last-gasp penalty.
Angers will play either Paris St Germain or Monaco, who face each other at the Parc des Princes on Wednesday.
Mangani opened the scoring in the 38th minute with an effort from close range to put the hosts, who were looking to reach their first Cup final since 1957, on the right track.
Former France international Briand had the chance to send the tie into extra time, but his penalty hit the post in the 89th minute.
Toko Ekambi scored two minutes into added time to end Guingamp's comeback hopes.
The final will be played at the Stade de France on May 27.