Despite going down 2-1 to Bayern Munich on Tuesday, Atletico advanced from the semi-finals 2-2 on aggregate courtesy of the away goals rule.
Xabi Alonso had given Bayern the lead in the first half but Griezmann struck eight minutes into the second to give Atletico the advantage after Thomas Muller had missed a penalty
Robert Lewandowski put Bayern back in front to set up a nervy finish but Atletico held on, despite seeing Fernando Torres fail to convert his spot-kick late in the tie.
Atletico will now contest a second final in three years, but Griezmann insists they must not get carried away.
"[Bayern] were better than us today, but we won the first leg and that's it. We scored an away goal, and that was most important," the France international told Antena 3.
"But we have to keep our feet on the ground."
Griezmann added: "We are happy. It was difficult, a complicated game.
"There was a great team in front of us, and we had to defend, so we had to try and get a goal and that's what we did."