Paulo Dybala struck the winner deep into injury time.
Allegri stated, “Football is vicious. We fought back for 2-2 and lost the Super Cup to Lazio in the last minute, tonight they were beaten at the last minute.
“It was a difficult game, because Lazio are in third place, they held out well physically and we had very limited options in attack.
“We played the game that we had to play and had to come away with the victory. I told the lads that a single incident would decide this match and I was right.
“It wasn't the best performance on a technical level and we gave the ball away too much, but we were alert on their counter-attacks, as they hurt us in the past with those.
“We had to be wide awake and stay in the running to the very end. I had no doubts on the character of my players and, as always, they were extraordinary.
“We had to stay in it and make the most of the incident that went our way. The goal means a lot for Dybala and the team."