Griezmann, 27, scored either side of half-time in a dominant performance by Atleti, only saying his aim was to "win more trophies" without being drawn on whether that would be with his current club or elsewhere.
"I need to just enjoy this moment, having won a title, and celebrate it with the fans", said Griezmann after picking up the man-of-the-match award for scoring the Spanish side's first two goals.
The former European champions then suffered a huge blow when Payet, an injury doubt before the final, left the pitch in tears.
But nothing could disguise Atletico's vast superiority, as it added to the Europa League trophies from 2010 and 2012.
A spirited Marseille team gave it their all, but Atletico were just too streetwise and also just too good.
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Atletico's build-up to this match has been overshadowed to some extent by constant speculation about his future, with the France striker being tipped to join Barcelona after the World Cup. The best team has won, the one with the most experience, it is a great team. "The Champions League is one thing and the Europa League another", said Simeone, who will serve a touchline ban on Wednesday after being sent to the stands against Arsenal in the semi-finals.
The game, at Lyon's Groupama Stadium, will be Marseille's first European final since 2003-04 when a side featuring Fabien Barthez and Didier Drogba finished runners-up to Valencia in the UEFA Cup.
"I hope he is happy with us".
In the final of the Europa League Spanish club with the score 3:0 defeated the French Marseille and won the trophy. The Frenchman's second goal of the night only reiterated what many had suspected leading up to the Europa League final - Atletico Madrid would outplay the Ligue 1 side even without fully exerting their might. "Firstly, we had the best chance to go ahead, and secondly an error from us meant we fell behind", said Garcia.
Atletico made a nervy start as Marseille flew out of the blocks but they took control in trademark fashion after the French outfit hit the self-destruct button.