Lionel Messi gave the hosts the lead from the penalty spot after 27 minutes following Layvin Kurzawa's clumsy challenge on Frenkie de Jong.
But the Mbappé show started five minutes later. The France international ran through the home defence to ram the ball past the Barcelona goalkeeper Marc-Andre ter Stegen.
Mbappé got his second midway through the second-half after Gerard Piqué failed to clear the ball.
And 70 minutes in there was further woe for Barcelona after Pedri's trip on Presnel Kimpembe.
Leandro Paredes floated the free-kick into the box and Moise Kean headed past Ter Stegen to make it 3-1.
Barcelona boss Ronald Koeman replaced Antoine Griezmann with Martin Braithwaite to add some muscle to the forward line in the hope of salvaging a second goal.
Clinical
But it was Griezmann's fellow France international forward Mbappé who had the last word.
The 22-year-old latched on to Julian Draxler's pass and curled the ball round Ter Stegen to complete his hat trick and set his side up for the second leg.
— Paris Saint-Germain (@PSG_English) February 16, 2021
"We're very happy," Mbappé told the Uefa website after the game. "It was a very important match for us. We wanted to come here and win. We did that in style. Tonight was magnificent but we haven't won anything yet."
Four years ago, Barcelona overcame a 4-0 defeat to PSG in a last-16 first leg by winning the second leg at home 6-1.
Then Messi was aided and abetted by the likes of Luis Suarez, Andres Iniesta and Neymar.
The latter now plies his trade in Paris while Barcelona have not replaced Iniesta's midfield guile nor Suarez's cunning in the penalty area.
In the prelude to the clash, PSG boss Mauricio Pochettino referred to Barcelona's comeback as history.
The 2021 PSG vintage have the chance to exorcise the ghosts on 10 March on 10 March at the Parc des Princes.
"It's huge," said PSG skipper Marquinhos. "We're going home with a win. We know the advantage it represents but we've learned from our mistakes. We'll stay humble. There are still 90 minutes to play but if we continue with this spirit and attitude, we can do great things."
Koeman conceded his side had been outclassed. "I could tell you lies but the fact is that at 4-1 down from the home leg, there are very few chances of going through," said the former Barcelona defender.
"This match shows us how much we have to improve, particularly at Champions League level."
