Juventus came back from a goal down in the second leg against Tottenham Hotspur at Wembley, behind a three-minute pair of goals by the Argentine duo Gonzalo Higuain and Paulo Dybala to advance past Spurs 4-3 on aggregate.
The victory sends Juventus into the quarter-finals of the Champions League for the third time in four years.
And Allegri, who managed Juventus to the Champions League final last season, is pleased to have continued his success in the tournament.
"First an foremost we're really happy that we won tonight and that we made it to the last eight,” Allegri said. "I've always said since I first came to Juventus that our ambition is to go as far as possible."
However, Juventus now sit in a top-eight that features the likes of runaway Premier League leaders Manchester City, and Real Madrid, the latter of who have won the tournament the last two seasons. They may also be joined in the last-eight by Barcelona and Bayern Munich, meaning Juventus may well find themselves matched up against one of those sides in the next round.
And Allegri is preaching caution, talking up the quality of the ‘favorites’ though promising his side will compete no matter the opposition.
"But you just have to look around you,” Allegri added. “We're in the last eight and there's some really good teams. People talk about Real Madrid and Barcelona, who are perhaps up there with the favorites, but that doesn't mean that we're not going to compete."
Juventus has advanced to the Champions League final in two of the last four seasons. In each of those appearances, they dispatched one of Real Madrid or Barcelona in the knockout stages, only to lose to the other in the final.