The European champions were cruising with a 4-1 lead early in the second half after a Lionel Messi brace and goals from Luis Suarez and Rafinha had cancelled out Ever Banega's opener.
But Unai Emery's men fought their way back and forced the game into extra time, only for Pedro to score a 115th-minute winner to secure a 5-4 win for Barca.
"It seems Barcelona are expected to win every title, but it is not always like this. People think Barca win automatically but that's not the case. We should be proud of these players," Luis Enrique told reporters.
"What we did has a lot of merit. So I am very proud of how the players managed to restart again when it was 4-4 and create new chances.
"When you're leading 4-1, the opponent has nothing to lose. Obviously we have to analyse what happened and the risk that Sevilla took. They managed to come back. It happens sometimes. They had a penalty and used the wings.
"It was the same last season in La Liga. We were leading 2-0 and finally they drew the game. Today it happened the same in a final, but we can't forget Sevilla's strength."
Luis Enrique said he felt the game was over in the 52nd minute, when Suarez struck to make it 4-1, but Jose Antonio Reyes, Kevin Gameiro and Yevhen Konoplyanka all scored to take the game into extra time in Tbilisi.
"We knew how tough our opponent would be. Games can start randomly, like this one did. I think it was a great game for the fans.
"By the third minute we were losing, but at the 50th minute it seemed done. It is always difficult when you're facing an opponent at such a high level, it is always difficult.
"We are happy for our fans and players. The players put in an important effort after Sevilla managed to equalise. We're satisfied and now it's time to recover."
