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.
How many Champions League goals has Kylian Mbappe scored?
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 United.
Mbappe has joined that pair in far fewer games, scoring 48 times in just 71 Champions League appearances to put himself in the top 10 of the all-time scorers list.
The 25-year-old will fancy his chances of climbing those ranks further between now and the end of the season too, with Real Madrid legend Alfredo Di Stefano just one ahead on 49 goals, and Arsenal's all-time leading scorer Thierry Henry sitting on 50 goals in the competition.
Mbappe's goals have come with two clubs, scoring six times in nine games with Monaco as they made it to the semi-finals before joining PSG and netting 42 times in 62 games for the reigning French champions.
The forward should have at least two more games to add to that tally before leaving PSG, with a semi-final against Borussia Dortmund to come after the German outfit saw off Atletico Madrid on Tuesday night.
Luis Enrique's side will be regarded as favourites to win that tie and progress to the final, where they would be bidding to lift the trophy for the first time in their history.
Who has scored the most Champions League goals ever?
At 25 years old, Mbappe still has plenty of time to significantly add to his Champions League tally, particularly if he joins the undisputed kings of Europe Real Madrid as expected this summer.
However, he has some way to go if he is to challenge the very top of the all-time Champions League top-scorers list, which is headed by his idol - and Real Madrid royalty - Cristiano Ronaldo.
The five-time winner of the competition netted 140 goals in 183 appearances from 2003 to 2022, including 21 in 59 games for Manchester United, 105 in just 101 outings for Madrid, and 14 times in 23 matches for Juventus.
Mbappe's former PSG teammate Lionel Messi is the only other player to have scored more than 100 Champions League goals, registering 129 times in 163 appearances in the competition - the third-best goals-per-game ratio of any player in the top 20 scorers, after only Erling Haaland and Alfredo Di Stefano.
The leading active scorer in the competition is Robert Lewandowski with 94 goals in 120 appearances, although he can no longer add to that tally this season after being knocked out by Mbappe's PSG on Tuesday.