Lionel Messi had put Barca ahead in the 27th minute from the penalty spot, but Mbappe brilliantly levelled five minutes late to end a thrilling first half all square.
Mbappe then capitalised on a loose ball to finish into a near-empty net after 65 minutes, and Everton loanee Moise Kean headed home unmarked five minutes later before Mbappe completed his treble five minutes from time to cap a superb turnaround for Mauricio Pochettino’s PSG.
The result means Barcelona will have to score at least three goals at the Parc des Princes in the second leg,
PSG headed to the Camp Nou without the injured Neymar and Angel Di Maria, while Barca were able to welcome back Gerard Pique following a three-month absence.
Antoine Griezmann and Mauro Icardi both went close at either end before the first flashpoint of the game saw Barca awarded a penalty.
Laywin Kurzawa accidentally clipped Frenkie De Jong’s heels with both chasing a long ball over the top, and referee Bjorn Kuipers pointed to the spot despite the furious protests of PSG players.
Messi stepped up and powered an unstoppable penalty high to Navas’ left to put Barca up, but their joy was short-lived.
Kurzawa responded well to conceding the spot-kick as he chased a long ball and pulled it back for Marco Verratti on the stretch: he flicked on to Mbappe, who shimmied away from Clement Lenglet with a glorious touch before blasting high past Marc-Andre ter Stegen from close range in a devastating display of technique, power and precision.
Ter Stegen then had to save well from Kurzawa, Greizmann dragged a shot just wide and Icardi headed into the side netting, but neither side could find a breakthrough before the half-time whistle.
PSG then went into cruise control after the break as Mbappe buried into the near-empty net 20 minutes after the restart, while Everton striker Kean headed home completely unmarked five minutes later to the joy of Pochettino.
But PSG weren’t done there as Barca left themselves exposed on the counter five minutes from time, with substitute Julian Draxler teeing Mbappe up for a sumptuous first-time finish into the top corner to cap a glorious night for PSG.