Mbappe had given PSG the lead at the Parc des Princes but a Leipzig side beaten 6-3 at Manchester City earlier in the competition did not crumble in the French capital.
Instead 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.
However, they were rescued by Mbappe and Messi. The Argentinian was set up by the France star for the equaliser and then coolly chipped in a 74th-minute penalty awarded for a foul on Mbappe.
Mauricio Pochettino's team, who also saw Mbappe miss from the spot in injury time, are now top of Group A at the halfway stage, a point above City, who won 5-1 at Club Brugge earlier.
Leipzig, beaten by PSG in the semi-finals in 2020, are bottom without a point to their name.
