The Spanish side opened the scoring after just four minutes, with Alvaro Morata equalising in the second half before the Uruguayan and the Brazilian secured the win
Barcelona completed a trophy treble in Luis Enrique's debut season in charge as they beat Juventus 3-1 in the Champions League final in Berlin on Saturday.
Luis Suarez produced his most crucial contribution of his short career at Camp Nou when he restored Barca's lead 22 minutes from time to effectively settle the game, after Alvaro Morata had cancelled out Ivan Rakitic's early opener.
A fifth European title looked set to be a more routine ask when Rakitic - the man to perform so capably in replacing the soon-to-depart Xavi this term - opened the scoring in the fourth minute with a trademark Barca goal, which saw nine of their 10 outfield players touch the ball.
However, while Barca enjoyed the better of the first half from there - threatening a Juve back line shorn of Giorgio Chiellini through injury - former Real Madrid man Morata struck early in the second period.
Having scored twice in the two-legged semi-final against his former club, Morata was afforded a simple finish after Marc-Andre ter Stegen had saved an initial effort from Carlos Tevez.
Suarez was the only Barca man not involved in Rakitic's opener, but the Uruguayan provided the crucial touch in his side's second, as he replicated Morata's finish when Lionel Messi's effort was spilled by Gianluigi Buffon.
Neymar had a header disallowed, via a handball, as Barca sought to put the game out of sight late on.
But it proved inconsequential in the end, as the Brazilian added a third with the last touch of the match to ensure that Luis Enrique's side repeated the treble-winning achievement of Pep Guardiola's Barca in his debut season of 2008-09.
The victory also provided Xavi with a successful send-off to his career with the Catalan giants before he joins Qatari club Al Saad.
European success continues to elude Juve, meanwhile, with the Serie A side coming up short in their attempt to win the continent's most prized trophy for the first time since 1996 and their own treble bid.