PSG won the match 4-2, but it was marred late on by Mbappe's first Ligue 1 dismissal, as he grew frustrated with some rough treatment.
The France international darted beyond Savanier on the break, before being cynically scythed down.
Mbappe reacted by running over to the Nimes man and pushing him to the ground, with both men subsequently sent from the field.
The young attacker will miss Ligue 1 games against Saint-Etienne, Rennes and Reims.
Savanier has been given a five-match ban for his dismissal.