The French forward, not long back from injury, was introduced in the 52nd minute with the score 1-0 after Mauro Icardi's early opener.
Mbappe headed home the second goal within ten minutes of his introduction, and teed up Mauro Icardi just two minutes later as PSG extended their lead.
And the former Monaco man completed his hat-trick with a quickfire double around the 80 minute mark, first sneaking a finish into the far corner after rounding the goalkeeper before latching on to Angel Di Maria's fabulous outside-of-the-foot through ball.
The 20-year-old is almost a year younger than Messi was when the Barcelona attacker scored his 15th goal.
Mbappe is the first substitute to score a Champions League hat-trick since Joseba Llorente for Villarreal in 2008, and is now among the ten highest French scorers in the competition.
The win keeps PSG top of Group A, five points clear of Real Madrid having won all three of their games so far.
The Ligue 1 leaders are yet to concede in this season's competition.