A brilliant first-half Paulo Dybala brace put Juve in control in Turin, with a header from Giorgio Chiellini providing the third goal after the interval.
Allegri's men are unbeaten in their last 18 continental home matches and the coach - who recently ended speculation linking him with Arsenal by signalling his intention to stay at Juve - hailed the performance of his side.
"Juventus have respect for Europe and above all are respected in Europe," Allegri told Mediaset Premium. "That is the greatest result of my three years here.
"It was a great performance, we played well technically both in attack and defence. It is the first step towards the semi-final, but the second leg will be a different matter.
"I cannot say what percentage our chances of progressing are. In Barcelona we are going to have aim to score goals."
Despite Juventus putting themselves in a superb position ahead of the return leg at Camp Nou, perfectionist Allegri believes his side could still improve.
"I am a dreamer, but also pragmatic and a realist," Allegri added. "We are in a growth process, I am happy with what the players are doing and we have made important steps forward in the last few years.
"We had several chances in the first half, moving the ball around and forcing them to shift, but we got a bit too greedy for an extra pass. That's something we need to work on and improve for the second leg.
"Juventus deserve credit, as everyone contributed to the defence by cutting off the passing channels, otherwise when the ball is at the feet of [Lionel] Messi, [Andres] Iniesta or [Luis] Suarez, you're going to have trouble.
"We were sharp, with good passing moves, but we could have done better and in some moments tried to make direct passes too often."