Barca looked on course to progress into the semi-finals when Raphinha scored first on the night to open up a 4-2 aggregate lead at Camp Nou, having won the first leg 3-2 at the Parc des Princes six days earlier.
However, a red card for Ronald Araujo before the half-hour mark proved costly for the Spanish giants, with PSG going on to score four unanswered goals that sent them through to the last four.
Ousmane Dembele scored against his former club to level things on the night, before Vitinha brought the aggregate scores all square 10 minutes into the second half.
Mbappe, who has been widely linked with a move to Barcelona's bitter rivals Real Madrid this summer, then took the game away from Xavi's men with two goals in the final half an hour.
The first came from the penalty spot as he coolly slotted past Marc-Andre ter Stegen after Joao Cancelo's rash challenge on Dembele inside the box, before the World Cup winner put PSG out of reach in the 89th minute by firing home from close range after Barca had failed to clear their lines.
Mbappe's brace at Camp Nou took him from 46 goals to 48 in the competition, moving clear of Eusebio and Filippo Inzaghi, and level with legendary former AC Milan duo Andriy Shevchenko and Zlatan Ibrahimovic.
Shevchenko needed 100 goals to reach that tally during his time with Dynamo Kiev, Milan and Chelsea, while Ibrahimovic played 124 times in the competition over spells at Ajax, Juventus, Inter Milan, Barcelona, AC Milan, PSG and Manchester
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