Neymar ensured the game was put to bed after he slotted the ball past Buffon at the end of stoppage time.
The win means Barcelona is the first European team to have won two trebles with the first coming in the 2008-09 season under pep Guardiola.
Luis Enrique become the second Barcelona manager to have won the treble in his first season.
The former player and B team boss Enrique took over as head coach last summer after the departure of Gerardo Martino.
Barcelona president Josep Bartomeu was pleased with the way the season ended for his team.
“It has been a tough season with ups and downs,” he added. “We took a lot of decisions and I am happy because they were the correct decisions.
“When they are correct at the end of the season it ends with La Liga, the Copa del Rey and the Champions League.
“I’m very happy for everybody and for all the people that have been supporting us.”
Juventus captain Gianluigi Buffon praised his team after they were beaten by ‘the best team in the world’ in the Champions League final.
“We didn’t manage to win, but not because we didn’t try. We gave everything we had,” said Buffon
“Barcelona are the best team in the world in my opinion, but we have proved that we could compete.
“We conceded early on, but we were always in the game.