Despite losing 2-1 to Barcelona at home, Atletico won the quarterfinal tie 3-2 on aggregate.
Griezmann said afterwards: "I'm very happy.
"Their goals came from two of our mistakes, and that's what you pay for in these kinds of matches. I lost the ball on the second one. I wasn't positioned correctly to pass it.
"Then, with our fans and the quality we have, we were able to score. We weren't comfortable on the ball. We didn't have the necessary composure to play our game, but, well, we're in the semifinals."
Atlético will find out their opponent later on Wednesday: Arsenal or Sporting CP: "It doesn't matter who we play as long as we're in good form until the end.
"It's been a very exciting and difficult tie against a fantastic team that plays very well. It's been incredibly tough, but we're through."