Diego Simeone's men surprisingly pipped Real Madrid and Barcelona to the Liga title last year and reached the Champions League final, only to lose in extra-time to their city rivals after conceding a last-minute equaliser.
France international Griezmann, who joined the club for a fee of around £19.8 million, feels that they will be competing on all fronts once again next year, despite losing the likes of Diego Costa.
"I am an ambitious player and the team will be battling to win every competition it is involved in," he said at his unveiling.
"But as the coach says, we'll take it one game at a time, and that means a lot of hard work and effort. I needed to take a leap in my career, to learn new things and I knew that the place to do that is here at Atletico.
"I hope to learn a lot from the technical team and also from my new team-mates.
"Atletico stated an interest quite some time ago, and when they called, they made their intentions very clear. This is a great club; they ended last season as League champions and runners-up in the Champions League.
"They didn't really need to say anything more to me.
"I don't feel under any pressure, I only want to enjoy my football and show my gratitude to the fans in every game, in every minute I play.
"I've come here to play, to grow as a player and help the team. We'll try to put in a season as good as last season's."