The France global gave his side the lead in Lyon with an cheeky finish after catastrophic defending from the French outfit.
Griezmann now has 29 goals this season, although it was Gabi's late strike that sealed the victory.
No French team has ever won the Europa League title.
Marseille was going for a second European trophy, 25 years after winning the Champions League, but ultimately had no answer to Griezmann's mobility and finishing. 'I hope he stays. Now we are in the UEFA Super Cup, so he could play four finals with us.
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. This [final] was a very important game, as it allowed us to keep up at the top level.
"It was a dream of mine to win this trophy with my teammates and with Simeone", the France forward said. They carried on anyway, and so did the game, with sections of the stadium cloaked in thick pyrotechnic smoke.
"The scoreline is harsh but the better team won".
Marseille skipper Dimitri Payet is comforted by Lucas Ocampos as he comes off injured in the first half.
With Payet's exit went any hope of them clawing a way back into the contest, which Griezmann and Gabi ensured was taken away from them completely in a 3-0 defeat, yet in his post-match media conference Garcia stood by his decision to start his hobbled star.
Click through the gallery above to see pictures of Atletico Madrid celebrating after beating Marseille in the Europa League final.
Goalkeeper Steve Mandanda opted to pass to Andre-Frank Zambo Anguissa, deep in midfield, rather than clear long.