Captain Loic Perrin was sent off for committing the nasty tackle in the 31st minute - in his final professional game after spending 13 years at the club.
Neymar had put PSG ahead in the 14th minute, converting the rebound after Mbappe's initial effort was blocked by goalkeeper Jessy Moulin.
Saint-Etienne had the first real chance of the match, hitting the post in the fifth minute through Denis Bouanga.
Mbappe was later seen on the sidelines with his right leg strapped and on crutches, but he was able to join in some of the celebrations with his team-mates.
The PSG squad celebrate winning the Coupe de France
Image credit: Getty Images
This is PSG's fifth Coupe de France in the last six years, and 13th overall.
The final was watched in person by 5,000 fans in the 80,000-seater Stade de France due to strict health protocols in the light of the coronavirus pandemic.
PSG, who are the Ligue 1 champions, next face Olympique Lyonnais in the League Cup final and have a Champions League quarter-final against Atalanta coming up next month.