Mbappe netted a hat-trick as the Parisians claimed a significant advantage ahead of the second leg in the French capital next month, leaving FC Barcelona on the brink of an early exit.
Messi had given Barca a 27th minute lead from the penalty spot after Frenkie de Jong was hauled down inside the box but Paris Saint Germain responded impressively, with Mbappe equalising in the 32nd minute.
The France international swept home in the 65th minute, before Moises Kean headed in for 3-1 five minutes later. Mbappe finished off a fine move to round out his hat-trick in the 85th minute.
“He is a top player. Great footballers write their history,” PSG boss Mauricio Pochettino said about Mbappe.
“They evolve match after match. With this hat-trick, Kylian will be in the spotlight but we had no doubts.
“He’s world-class. No one can discover Mbappe tonight. He’s a fantastic player and he’s very happy in our club. He said it, I said it too.”
Elsewhere in the Champions League on Tuesday, Liverpool put aside their domestic struggles to claim an important 2-0 ‘away’ win over Leipzig in Budapest.
In a tight match, Liverpool capitalised on two defensive mistakes to score in quick succession from Mohamed Salah and Sadio Mane.
“Very, very pleased with the performance obviously,” Reds boss Jurgen Klopp said. “It led to a result, I think to a deserved result.
“It was a game where you dont expect big advantages for one or the other team because of the nature of both teams, because both are really well organised, both defend really well, both are used to causing the opposition side problems in the way they want to build the game.
“I thought that worked out slightly better for us tonight.”