"The truth is this was a dream for me, to win a trophy with Atletico, with [Diego Simeone] and my friends and teammates", he is quoted as saying by ESPN.
Griezmann's two goals on Wednesday took his tally for the season to 29, while he has also assisted a further 15 strikes in 48 appearances across all competitions for the Rojiblancos.
Marseille was going for a second European trophy, 25 years after winning the Champions League, but had no answer to Griezmann's mobility and finishing.
But the France worldwide is linked with a move to Barcelona in the close-season despite being contracted until 2022. "I want to enjoy the moment".
"We hoped that he'd be at 100 percent and we gave him a fitness test this morning".
"Simeone made me grow personally and professionally; I'm delighted with him and I owe him a lot, I know, so I'm very fond of him".
"I hope he has the same career because he deserves it and takes this team where he wants, at least among the top sides in the world".
Marseille began with urgency, but striker Valere Germain shot over from playmaker Dimitri Payet's slick pass after three minutes.
"We let our chance go in the first half. You can't fall behind against a team like Atletico", Marseille coach Rudi Garcia said. Captain Gabi capped off Atletico's victory with a fine finish in the last minute after being set up by Koke.
Griezmann got two chances, one either side of halftime, and made no mistake as the Madrid team took advantage of some poor defending in the first half to send the French man in one on one with the Marseille keeper and he finished calming.
Atletico are celebrating winning the Europa League final last night, but Lacazette says: "If it was unusual to play against Atlético Madrid that I could join a year ago? No".
Payet was a key figure in France's run to the Euro 2016 final, although he has not featured for the national team since a 2-1 win over Belarus on October 10.