A corner from the right by Griezmann bounced up into Perisic's hand and after initially giving a goal kick, Argentinian referee Nestor Pitana reversed that decision and awarded a penalty - duly converted by Griezmann - after consulting the VAR system.
Hugo Lloris made a blunder in his own goal to help ease the pain of scoring an own goal for Mandzukic.
It's the first ever final for the Croatians, while France is there for the third time and goes into the match as the betting favourite.
"It's so attractive, so wonderful", Deschamps told France's TF1.
"I don't really realise yet what it is".
France won their first World Cup championship 20 years ago in 1998.
Croatia tried to gain a foothold in the game but France held firm, using the pace of Mbappe on the break to keep their opponents pushed back.
France striker Antoine Griezmann: "I do not know where I am!"
Croatia's shirtless fans rightly saluted their players as torrential rain lashed down on the presentation ceremony and coach Zlatko Dalic had few regrets. This "golden generation" led by Luka Modric, will not be overawed by the fact they are simply in the final; Mario Mandzukic and Ivan Rakitic have already scored goals in past UEFA Champions League finals.
As a thunderstorm erupted in the sky over Moscow, Griezmann stroked home the penalty, his fourth goal of the competition and his third from a spot-kick.
Four pitch invaders disrupted the game in the 52nd minute for about a minute before being dragged away by security and police.
In the 59th, a run from Mbappe started a play that ended up with Pogba on the edge of the penalty area.
Mbappe, 19, netted the goal in the 65th minute.
An bad blunder by Lloris revived Croatia, as the French keeper tried to dribble around Mandzukic only for the striker to tap the ball straight into an unguarded net. That's double the previous World Cup record of six.
Croatia has not beaten France in five games since being accepted as a FIFA member federation in 1992.
Les Bleus had a touch of luck - and some controversy - in the 18th minute when they had yet have a shot on target. The team's current head coach Deschamps was the captain of the World Cup-winning squad back in 1998.