Griezmann got his first mid way through the first half and his second just after the pause. Gabi added the gloss in the closing stages against a willing but ultimately limited Mrseille side who had been hampered by the loss of their skipper and playmaker Dimitri Payet in the early stages of the final in Lyon
Griezmann, who was brought up 70 kilometres from Lyon, said: "This is a reward for me after I left home aged 14, for all the work I have put in, the suffering."
Atletico's triumph is their third in the competition since it emerged from the Uefa Cup in 2010.
Diego Simeone has steered Atletico to all the of their recent European successes and the Argentine said the win not only vindicated the squad's hard work but also confirmed Griezmann as one of Europe's most coveted strikers.
"This is what Griezmann has been doing year after year," said Simeone. "I hope he is happy with us. I have no doubt that there are certain things that could make it more likely that he stays.
"He has played three finals with us, has won two, and so that shows that he is not far away from his objective of regularly playing finals. We are not far away in sporting terms from teams who are more powerful than us."
Marseille had been hoping to lift their first European trophy since their victory over AC Milan in the Champions League final in 1993.
The French side had the first real opportunity of the match after four minutes. Payet slipped the ball through to Valère Germain in the penalty area. But the 28-year-old Frenchman fluffed his shot.
Seventeen minutes later, Griezmann showed his compatriot the essence of composure. Marseille goalkeeper Steve Mandanda opted to pass to Andre-Frank Zambo Anguissa in midfield, rather than clear the ball upfield.
Zambo Anguissa failed to control the ball properly, Gabi pounced on it and his first touch sent Griezmann through. The 27-year-old waited for Mandanda to move and he slotted the ball past him into the net.
Griezmann, unlike Germain, will be in the France squad for the 2018 World Cup in Russia.
"We let our chance go in the first half," said Marseille coach Rudi Garcia. "Firstly, we had the best chance to go ahead and secondly an error from us meant we fell behind. The scoreline is harsh but the better team won."