Ferran Torres scored in the first-half as Barca won the second-leg 1-0 at the Metropolitano to reach the final 5-4 on aggregate.
Flick said afterwards: "We were very brave. We only scored one goal, but it was enough to get to the final. Everyone should be proud.
"We know it's very difficult to win here. Atlético is a great team with great players. But we've done well; we've improved in the last month.
"It's not easy to win here. But the defence worked very well. I'm happy."
Reaching the final, where they will meet Real Madrid, keeps Barca's hopes of a Treble alive.
Flick continued: "It's allowed to dream, but we have to stay focused and focused on what we want. We'll arrive at three or four, then we have to sleep and get back to training and recover. We're facing Betis.
"The next few weeks won't be easy, but we have a fantastic team. The club and the fans can be proud of the team."
He added, "I don't want to talk about the final. I'm focused on Betis. But it's fantastic to play El Clásico, but you have to play a lot of games. There are a lot of important things. Now isn't one of them."