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.
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.
For PSG the win was achieved without Neymar, sidelined by an adductor muscle injury.
In any case, the world's most expensive player has not really been himself this season, his one goal so far coming from the penalty spot.
While Messi has been settling in to his new surroundings, Mbappe has been the main man for PSG.
Glitz and glamour
Those two were joined in the Paris attack by Julian Draxler, making a rare start with Angel di Maria suspended and Mauro Icardi missing for family reasons amid reports of relationship troubles with his wife Wanda Nara, who is also his agent and an Instagram celebrity.
Adding to the glitz and glamour of the Parc des Princes was Ronaldinho, the former PSG player and later a team-mate of Messi's at Barcelona. The Brazilian embraced Messi during the warm-up before performing a lap of honour.
Ronaldinho, for a brief period arguably the best player in the world, could not fail to have been impressed by Mbappe, who looked in the mood as he gave PSG the lead nine minutes in.
As Silva appealed in vain for a free-kick on the edge of the Paris box, the home side broke quickly and Draxler released Mbappe just inside the opposition half.
The France striker sped towards goal before unleashing a low shot past Peter Gulacsi for his sixth of the season, his first in the Champions League.
Yet Leipzig, who have been struggling for consistency in the Bundesliga under new coach Jesse Marsch, were a threat.
Silva hit the post with a shot on the turn, and moments later the German side were level in the 28th minute, Angelino with a low ball across goal for Portugal international Silva to turn it in.
Angelino was again the provider when Leipzig took the lead, crossing for French international Mukiele to stab the ball past Keylor Navas in the 57th minute.
PSG were struggling. They still don't look quite like the fully-rounded team that can dominate Europe, but their superstar individuals stepped forward when it mattered.
Mbappe set up Messi to equalise midway through the second half, the Argentine following in to make sure when his initial effort hit the post.
That was his second PSG goal following his stunning effort in the win over City last month.
He wasn't finished there, cheekily beating Gulacsi from the spot for the winner after the brilliant Mbappe had been fouled by Mohamed Simakan.
The Argentine might even have had a hat-trick when another penalty was awarded in stoppage time for a foul on Achaf Hakimi after a VAR review.
However this time Mbappe stepped up and blazed the kick over the bar.
Standout matches today