The German side snatched the 2-1 come-from-behind triumph when Javi Martinez netted an extra-time winner.
Flick acknowledged his players were slow to start at the Puskas Arena in Budapest.
"It took the first 15 minutes for us to get into the game," he was quoted as saying on uefa.com.
"It was great for me to see the team try to pull through with their mentality – and succeed. In the end, my players went through the pain barrier to win," Bayern's manager added.
Sevilla substitute Youssef En-Nesyri was denied late in normal time by a fine save from Manuel Neuer, who subsequently diverted another shot from the Moroccan against the post in the second minute of extra time.
Martinez, another substitute, then provided the decisive moment in the 104th minute, heading past Yassine Bounou after the goalkeeper had parried an Alaba shot, as Hansi Flick's treble-winners secured another trophy to add to their haul.
Sevilla coach Julen Lopetegui was grateful to have matched "the best team in Europe" for much of the game, but still noted the pain of the loss.
He said: "It always hurts to lose, especially in a final and especially in one we had chances to win.
"We had to suffer and we were able to hurt them, and they got the winner when it least looked likely.
"We've only been back in training for 12 days and we were able to go toe to toe with the best team in Europe and take them to extra time, which is more than most."