Atletico blew a 2-0 home lead as Barca struck four goals inside the final 20 minutes.
Flick said afterwards: "I think we played well in the first half. After conceding two goals, we showed great mentality. We fought and created incredible chances. I'm very happy for the team, the club, and the fans.
"Today was the day we had to win after three games. The game was incredible. We're very happy. After the second goal, the reaction was incredible. We scored four goals in 20 minutes
"I'm happy and thrilled for this team. They haven't thrown in the towel. The reaction has been incredible; that's how we want to perform on the pitch. I'm very, very proud.
"We play the way we train, and they're doing very well. We're very confident in how we want to play, and when this team makes a mistake, they capitalize on it. They're a fantastic team, with passion."
Flick added, "Being two goals behind against Atlético and turning the score around in 20 minutes is very important. It shows that the team believes in its strengths. The break is good for everyone. I'm happy that the players won't be playing for another two weeks.
"We had a lot of possession, even in the first half. We made some mistakes and gave Atlético a chance to score two goals. We believed in our style. There's a great rapport on the pitch and in the locker room; everyone is working in the same direction; we're a family. What I feel as a coach is incredible. I really enjoy working for this club."