Defending champions PSG, who have appeared in the last six French Cup finals, winning five, controlled possession, but Angers had their chances as Ibrahim Amadou fired narrowly wide and Jimmy Cabot saw his effort brilliantly saved by Sergio Rico in the home goal.
PSG hit the front inside nine minutes with an excellent team goal that was finished by Icardi from Julian Draxler's pin-point through ball. There was some good fortune about the second as Neymar's shot from close-range hit goalkeeper Ludovic Butelle, before rebounding onto Manceau and back into the net.
Neymar headed in from close range before a brilliant run in the box from Angel Di Maria put a second on a plate for Icardi, who then added another late on for his fifth goal in the last three games.
Lyon dominated the first half against visitors Monaco but were held scoreless, and the game slipped from their grasp early in the second period.
Lyon's Maxwel Cornet had a goal correctly ruled out for offside, before Sinaly Diomande was sent off for a second bookable offense for his role in giving away a penalty.
Wissam Ben Yedder converted from the spot for Monaco and set up Kevin Volland for a thunderous strike to make it 2-0.
Lyon had earlier lost coach Rudi Garcia to a red card after a show of dissent to referee Stephanie Frappart.
Montpellier and fourth division Rumilly-Vallieres, who stunned second-tier Toulouse on Tuesday, are the other sides to have sealed a semifinal place.