The France international netted in both halves as Los Rojiblancos carved out a 3-0 win at the Groupama Stadium in Lyon on Wednesday night.
Atletico have now lifted the trophy three times since 2010, but this latest success was Griezmann's first major honour in his four years in the Spanish capital.
Amid fresh talk of a switch to Barcelona being close to completed, the 27-year-old was only interested in thanking manager Diego Simeone and his teammates after the European triumph in his native France.
"I think now is not the moment to talk about my future, I just want to enjoy myself after my performance, after winning a trophy and I want to celebrate with the fans," he is quoted as saying by Sky Sports News.
"Simeone has taught me a lot, he made me become a better player and I hope I can keep improving every year, in every game. I owe a lot to Simeone and my teammates."
Griezmann's final Atletico game could be Sunday's meeting with Eibar at the Wanda Metropolitano.