A brace from Paulo Dybala and another from Giorgio Chiellini in Turin last week left the Bianconeri firmly in control ahead of Wednesday's quarter-final second-leg tie at the Nou Camp.
"They are an Italian club and defend very well. From a defensive point of view, from the whole team, it was an outstanding night for them", he added. "It was a hard result to turn around, although we tried until the end".
Luis Enrique felt his side did enough to win on the night but noted it was always going to be tough to produce a repeat of their miraculous comeback performance in the last round against Paris Saint-Germain, when three goals in the last seven minutes saw Barca through.
"The self-belief of the team has grown a lot", said Allegri.More news: Texas House approves major school finance fix; Senate's plan uncertain
"My players were extraordinary tonight".
Messi could have boosted Barcelona's hopes in the 19th minute when a long buildup ended with Luis Suarez setting him up near the penalty spot, only for Messi to fire wide.
Captain Gianluigi Buffon hopes the comprehensive win over five-times European champions Barcelona is a sign his side are ready to take the next step.
Barcelona earned corner after corner - 13 in all - but could produce no joy in the late stages and for all their attempts finished with only a single shot on target, their fewest in a home Champions League game in the past seven years.
"(Barcelona) never really troubled the goalkeeper.More news: United Airlines can't say if video has hit sales
Barca coach Luis Enrique had said before the game that if hia team managed to strike early then the noise of the Nou Camp would suck the other goals in.
Barca will be without Neymar at the Bernabeu through suspension and the Catalans will need Lionel Messi back in top form as the Argentine was off target with the only two clear chances the hosts created on Wednesday.
And the only negative news of the night for the Bianconeri was a yellow card for Sami Khedira that will rule the German worldwide out of the first leg of their semifinal. Having now avenged their 2015 Champions League final loss to Barcelona, Juventus also leads the Italian league and has lined up an Italian Cup final against Lazio.
- Buffon's 2016-17 Champions League campaign: Nine games seven clean sheets, two goals conceded.
After Barcelona's Champions League exit, they are all set to face their rivals Real Madrid on Sunday in a potential title-deciding Clasico.More news: Kompany scores as Manchester City climb to third with Southampton victory
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