Quick thinking by Mbappe to disposes Axel Disasi in his own penalty area, as the Monaco defender let a short pass from goalkeeper Radoslaw Majecki slip under his feet, set up a 19th-minute opener for Mauro Icardi.
Mbappe took the ball away and, displaying quick feet, passed square for Icardi to tuck the ball home from close range.
With teammate Neymar suspended and watching from the empty stands, Mbappe was able to dominate the contest with his talent and pace and added the second goal in the 81st minute.
Angel Di Maria set him up with a short through-ball and the World Cup winner lifted his finish over Majecki to make sure of the outcome.
Mbappe's goal came a minute after he had hit the crossbar with a clever chip from outside the penalty area having spotted Majecki off his line.
It was one of few chances in a largely dour final, played behind closed doors, that only came alive in the last 15 minutes.
PSG retained their title and extended to 14 their record number of wins in the cup. Monaco had won five previous times but last in 1991. They were last in the final in 2010 when PSG also beat them.
"We work every day to experience this type of emotion, reward and recognition. The work of a whole group, the staff, and the supporters who could not be here. We think about them a lot, this title is also for them," Mbappe said.
"When you play for PSG, one of the biggest clubs in the world, the biggest club in the country, every title counts to go down in history. We want to be part of this story, it's a great step today."
PSG defender Alessandro Florenzi had a powerful shot tipped over by Majecki in the first half while Monaco created the better opportunities after the break.
But any hopes of a comeback were swiftly ended by the pace and vision of Mbappe as he settled the outcome.
Both clubs now turn their attention to the end of the Ligue 1 season on Sunday where they hold out hopes of winning the title. PSG are second, one point behind leaders Lille, while Monaco are three points adrift.