The Rossoneri had admitted they would need Lionel Messi to have a off night but the Argentinian forward single-handedly cancelled out Milan's two-goal advantage inside 39 minutes at the Nou Camp. And Milan had no subsequent answer to goals three and four from David Villa and Jordi Alba, with Barca wrapping up a 4-2 aggregate win to reach the quarter-finals.
Milan's exit may be galling for Allegri but the 45-year-old is now free to focus on chasing down Napoli and second place in Serie A, and he told acmilan.com: "We did well in the first leg but Barcelona's potential hasn't changed. They're the best team in the world with that trio of Messi, (Andres) Iniesta and Xavi."
He added: "Milan played a worse game than we did in the first leg and in the first 30 minutes of the match, they didn't allow us to play.
"We made a lot of technical mistakes and then when they slowed down, we had a chance through (M'Baye) Niang. If Niang had scored that chance we'd be here talking about a completely different result.
"Sometimes, matches are decided by centimetres and this time it was the post that kept the ball out. Myself and the players have nothing to regret, though.
"Barcelona deserved the end result but getting through to the next round would have been a big achievement for us. I'm very sad for the players, for the fans and for the club as well."
Rossoneri defender Ignazio Abate, however, did not try quite as hard to hide his disappointment. The full-back is now eager to cure his European hangover with victory over Sunday's opponents Palermo, who will be eager to impress new coach Giuseppe Sannino by reigning in a five-point gap on 17th-placed Genoa.
"There's a bit of bitterness as we could have managed things better in the middle of the field," he said. "It's a shame as we had the chance to do them some damage. This season we had a real chance to go further in this competition by knocking them out.
"But the main objective is the league and we can't drop our guard, Palermo have changed their coach and Sannino will play with 11 men in defence. There are about five teams within a five-point distance of each other and we can't give up."
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