Jules Kounde struck an extra-time screamer for the winner, as Barca triumphed 3-2 on the evening in Seville.
Real Madrid also had three players sent off at the end, with Toni Rudiger and Lucas Vazquez shown red over protests from the bench, with Jude Bellingham also sent off for remonstrating with match officials.
Flick declared afterwards: "I'm very proud of my team. It's incredible that they always have a response and fight for everything, that they never give up. Everyone is proud of this team and they deserve it.
"When I saw the celebration, it was incredible, and they deserve it. I'm very happy. It's a great victory. Koundé saved two chances for Real Madrid and scored the goal. It was incredible."
"We're all doing a great job - not just the coach"
Asked about his own influence, Flick stated: "I'm here to create an atmosphere at the club, in the locker room, so they can enjoy football and play at the highest level. They want them to see that we believe in them and that we'll be together until the end of the season. We're all doing a great job, not just the coach. And of course the players, with a fantastic mentality. It was great to watch the comeback from the sidelines. It was a perfect night.
"When you take on a job like this, you have to think positively. I was surprised by how well they welcomed me from day one, and I've always felt very good and very happy. We're all doing our best to succeed. Everyone is striving to improve more and more."
The Copa triumph means a Quadruple is achievable, given they've also won the Super Cup.
On reaching the final of the Champions League, Flick said: "There are two games left to reach that point in the Champions League, but today was fantastic. It was a great atmosphere, both inside and out. I'm very happy to be able to celebrate a huge victory against Real Madrid with them, and it gives us confidence to win more."