France international Griezmann, who has been heavily linked with a close-season move to Barcelona, scored twice and played an influential role in Gabi's 89th-minute goal as Atletico overcame Marseille 3-0 in Lyon.
The Supercopa de Espana was the only title the 27-year-old had got his hands on since his arrival from Real Sociedad and the in-demand forward admitted winning even more is his priority.
"I left home at 14 years old because I wanted to win titles. This is the second after the Supercopa de Espana and hopefully there can be more," Griezmann told beIN SPORTS.
"Marseille are an attacking team. They work very hard. We did our thing, defended well, were a hard team and took advantage of an error."
Barca are said to be keen to activate a reported €100 million release clause in Griezmann's contract, but Atleti general manager Clemente Villaverde insists the Rojiblancos are desperate to retain him.
"Antoine is doing great things, he made that clear [on Wednesday]," Villaverde said.
"He has a contract and we want him to stay. We are making every effort."
Atleti right-back Juanfran expressed confidence that Griezmann will stay put.
"I am sure we will be the same and I'm sure we will return to another final to try to win again," he said.