Mbappe gave the hosts the lead in the 14th minute, threading the ball between Nice keeper Marcin Bulka's legs after a skilful one-two with Fabian Ruiz.
PSG doubled their lead in the 33rd when Bulka sent a clearance to winger Ousmane Dembele, with the ball spinning to Ruiz, who calmly slotted into the open net.
Nice pulled a goal back four minutes later when Jeremie Boga embarked on a splendid run into the box and the ball found its way to Gaetan Laborde, who fired home on the half-volley.
Lucas Beraldo, however, wrapped up the win for PSG when he headed home on the hour mark.
There was a long delay minutes after the goal as Bulka took a knee to the nose which required treatment and a change of shirt for the keeper.
PSG will face Stade Rennais in the semi-final on April 3rd, while Lyon and Valenciennes will contest the other semi the day before.
