Mbappe netted a fine equaliser in the 32nd minute, shortly after Lionel Messi had put Barca ahead with a penalty.
The France forward subsequently struck again midway through the second half, Moise Kean added a header five minutes later and Mbappe completed his treble with an emphatic curling finish with five minutes remaining.
Griezmann, who won the 2018 World Cup alongside Mbappe for France, told RMC after the game: “We all need to enjoy Kylian, because PSG have a player who will be of the class of Leo and Cristiano.”
Barca coach Ronald Koeman admitted that the best team won on the night, with the BBC quoting him as saying: “Surely the result reflects how superior they have been. In the second half we had problems defensively. They proved physically to be very superior to us.
“We have to admit that they have been superior. They have shown that they have a more complete team than us. We have to accept it, improve things. We know that this can happen because we play against a great team. physical team, with experience ... for many things that is ahead of us.
“A 1-4 defeat is normally very difficult. I can lie to you but losing at home 1-4 there are very few options.
“The game has shown us that we lack things to be at the best level, especially at the Champions League level.”