The win means Barcelona become only the third Spanish side in history to repeat a domestic league and cup double in consecutive football seasons, although they had to dig deep after playing for 50 minutes with a player less as Javier Mascherano was sent off 10 minutes from the end of the first half, reports Xinhua.
Not until Sevilla's Ever Banega also saw red in the last minute of normal time did Barcelona step forward from their solid second half defensive display and Lionel Messi set up Jordi Alba and Neymar in extra time for the decisive goals to round off a season in which Barcelona also won the European Supercup and the World Club Championship.
"It has been a spectacular end to the season. The team has given 10 out of 10," said coach Enrique.
"Sevilla pressured hard, but were unable to create clear chances.
"All Barca fans should feel proud... This side could not show more character and personality. Everyone gave a little bit more than they had inside them and they knew how to close spaces and compete. It makes me proud and means I will enjoy the summer," added Enrique.
Enrique said the affection of the fans meant he was very happy in Barcelona, saying the season had been "easier" than his first campaign, in which they won the treble.
"The players adapted to my way of working. The dynamic of the team was more positive and if we didn't win everything this year, it was not because of the players, but because this is sport," he concluded.