Griezmann struck in extra-time as Atletico won 4-2 on Thursday night.
He said: "We know that it is a team that always fights, attacks and those at the back played a great game. In the end we were able to score that last goal which gave us a bit of relaxation.
"It's a great victory but we have to keep going like this because it's nothing.
"I will now go home and see my goal. Very happy with the team's effort, we have ended up tired, we have a few days to recover, but the truth is that we are very happy.
"Revenge no. We wanted to play the Cup at home, which we had wanted to win for a long time. There was a great atmosphere, it gave the team wings to fight and to play a game like that.
"Luckily we have beaten Madrid twice, it is true that we are very strong here. Now we have another important game on Monday and then we'll see. We are still in January and we have to keep fighting."