"It was a dream of mine to win this trophy with my teammates and with Simeone", the France forward said.
Griezmann now has 29 goals this season, although it was Gabi's late strike that sealed the victory.
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.
The 27-year-old France global has been heavily linked with a move away from the Wanda Metropolitano this summer and showed his class as Atleti ran out comfortable 3-0 winners in Lyon.
Marseille did not create enough to break down one of Europe's toughest defences and their chances of causing an upset took a blow with Dimitri Payet limping off injured in the first half. "I want to enjoy the moment".
The striker missed a penalty in the 2016 Champions League final against Real Madrid which his side lost in a shootout and that made him more determined to perform against Marseille. "Simeone has taught me a lot".
"We let our chance go in the first half".
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"I am proud of the season Marseille have had, of their fans who on their own nearly filled the whole stadium, and of (coach) Rudi Garcia and what he has done with that team".
"I left home when I was 14 wanting to win titles", he said after the match.
"I hope he is happy with us".
The France global produced an ice-cool finish beyond Marseille goalkeeper Steve Mandanda to put Atletico in charge and he started and finished an exquisite move which effectively killed off the French side early in the second half.
Atletico are celebrating winning the Europa League final last night, but Lacazette says: "If it was odd to play against Atlético Madrid that I could join a year ago? No".
The winners of the Europa League are awarded direct qualification for next season's Champions League group stage.
Things got worse for Marseille when Payet, their captain and most influential player who had been struggling with a hamstring problem ahead of the game, came off crying just after the half-hour.