Goals from Neymar and Angel Di Maria had given the French champions a comfortable 2-0 lead in a first half that saw Mbappe pick up a yellow card for dissent.
The young attacker appeared to be angry with the amount of fouls the hosts had been committing and had to be calmed down by Neymar.
The second half was more topsy-turvy, with PSG's lead having been cancelled out by a strike from Antonin Bobichon and a penalty from Savanier.
Mbappe himself was the one to restore Les Parisiens' lead with a fine strike before laying on another for Edinson Cavani in injury time, but the France international saw red in both senses shortly after.
A poor tackle from Savanier prompted outrage from the 19-year-old, who shoved over the midfielder upon getting back to his feet.
Both players were then given their marching orders.
PSG goalkeeper Alphonse Areola, preferred to Gianluigi Buffon on Saturday, also tried to calm down Mbappe, but he too was given a shove for his efforts.
Regardless, the champions picked up another three points to extend their lead at the top of Ligue 1 and retain their winning streak since the start of the 2018-19 season.
Paris Saint-Germain return to action against Saint-Etienne on September 14 following the conclusion of the upcoming international break.