Alvaro Morata returned to haunt his boyhood club, scoring the 57th-minute equaliser which was enough to send Juventus to Berlin with a 3-2 aggregate win after a 1-1 draw on the night.
“I have to congratulate the boys, it is a big night for them and for all Italian football,” Allegri said on the Juventus website. “I realise that this season is really great.”
Madrid dominated the second leg on home turf, but could only produce a Cristiano Ronaldo penalty and a host of near-misses, allowing Juve to profit.
“We know we have quality, but these two legs of the semi-final were really special,” Allegri said. “We had chances, it was a great match between two teams with good technique.”
Carlos Tevez, key to Juve holding the ball up on those occasions they managed to wrestle it away from Madrid, added: “Now we must use this moment. My thought is that this team deserved this final and we go to Berlin to play the match against Barcelona.
“The greatness of this team is as a group. We have shown throughout the game that we are here to do great things.”