Despite a second-leg defeat at Tottenham on Wednesday night, Atletico reached the final eight 7-5 on aggregate.
Griezmann said afterwards: "Very happy to have reached the quarterfinals, and for the fans who came to see us. There are things to improve, but we're going to celebrate today.
"It was hard for us to get into the game and find solutions. These things happen to us and we have to fix them."
On Atletico goalkeeper Juan Musso, the veteran said: "Musso has played some fantastic matches, he's saved us many times. We have two great goalkeepers and we have to celebrate them."
"Last year in March we lost all the titles, it was a blow and it was hard for me to come back, but here I am."
On facing Barcelonain the quarters, Griezmann added: "They're going to be two difficult matches. We can't play like we did in the second leg there. There will be time to prepare well."