Mbappe had given PSG the lead at the Parc des Princes but a Leipzig side did not crumble in the French capital, Goals from Andre Silva and Nordi Mukiele had them in front and at that point PSG looked as if they might pay the price for another occasionally disjointed display.
But then Mbappe set up Messi for the equalizer before the Argentine coolly chipped in a 74th-minute penalty awarded for a foul on the France forward.
In contrast Leipzig, beaten by PSG in the semi-finals in 2020, are bottom without a point.
PSG's win was achieved without Neymar, sidelined by an adductor muscle injury.
