However, Barca's ability to come back from horror away fixtures will firmly be in the mind of Juventus when they play the second leg in Barcelona next week.
It was sweet revenge for Juventus who lost to the Catalans in the Champions league final of 2015 the last time both giants clashed. "The feelings as manager are lamentable, and it's hard to explain. Juventus was superior to us in every way", Barcelona coach Luis Enriquesaid.
"I can not say what percentage our chances of progressing are".
"These are the players who really have to feed the front three and they're not getting the ball to them quick enough".
"I'm an optimistic person".
"I am a dreamer, but also pragmatic and a realist", Allegri added.
"In the first half the (Barca) players were determined, but we made the same mistakes from Paris, and that's a problem".
"The second half was much better, but I can't get the first half out of my head".
Despite Juventus placing themselves in a brilliant spot ahead of the return leg at Camp Nou, perfectionist Allegri considers his side could still enhance.
Juventus was aggressive from the start and nearly scored in the third minute when Gonzalo Higuain's header from a free kick taken by Miralem Pjanic was dropped by Marc-Andre ter Stegen, but the Barcelona goalkeeper recovered in time.
The 23-year-old struck twice with his lethal left foot inside the first 22 minutes of the UEFA Champions League quarter-final first leg in Turin.
However, after playing a starring role in his side's 3-0 Champions League win over Barcelona on Tuesday, he told Spanish TV channel Mega: "I'm very happy at Juventus and I'm very close to signing a new contract".
Two first-half goals from Argentinian striker Paulo Dybala put Juve in a commanding position by the interval, before a header from veteran centre-back Giorgio Chiellini sealed a humiliating result for the visitors at the Juventus Stadium.