Neymar rolled in a spot-kick to put PSG 2-0 up at the break on a hot night at the Parc des Princes after Falaye Sacko's own goal had put them ahead.
The Brazilian then headed in his second and PSG's third shortly after the interval, taking his tally for the season to five goals already after just three games.
Wahbi Khazri pulled one back for Montpellier before Mbappe made it 4-1 and new signing Renato Sanches came off the bench to get the fifth late on.
Enzo Tchato notched another consolation for the visitors at the death but PSG have now recorded three wins out of three this season under new coach Christophe Galtier, scoring 14 goals in the process.
Far from his best level, Mbappe still found the net with a flick from a corner in the 69th.
"Physically, we knew his lack of game time in pre-season would cost him," Galtier told broadcaster Canal Plus when asked about Mbappe's performance and attitude, with the 23-year-old appearing frustrated at times on the night.
"He is a competitor, he wants to be good, and he wants to be good quickly."
It looked like it was game over when Neymar added a second soon after the break, heading home a rebound as he reacted quickest to the loose ball.
Neymar was denied his hat-trick when he had an effort disallowed for offside, but PSG extended their advantage through Sanches on his debut following his move from Lille.
Tchato pulled another goal back for Montpellier with virtually the last kick of the game, but it was a comfortable victory for the home side.
PSG are expected to make more signings before the transfer window closes and have been strongly linked with Napoli's Spanish international midfielder Fabian Ruiz.
Earlier, Monaco came from behind to claim a point in a 1-1 home draw with Stade Rennais, a match in which they were reduced to 10 men early and missed a penalty.
Youssouf Fofana was shown a straight red card inside 15 minutes for a stamp on Rennes’ Martin Terrier, but Monaco won a first half spot-kick that was missed by Axel Disasi as Rennes goalkeeper Steve Mandanda made the save.
Gaetan Laborde opened the scoring for Rennes early in the second period, but an error from replacement goalkeeper Dogan Alemdar, on for Mandanda at halftime and who dallied on the ball to allow Breel Embolo to nick it from him and score his first Monaco goal, saw the points shared.