Federico Baschirotto, 28, opened the score for the visitors after the fault of Strahinja Pavlovic De Cremonese could nod from close home.
Federico Bonazzoli recovered their lead just after the hour with an early goal of the seasonal candidate and hit an acrobatic overhead kick past Mike Maignan.
After the game, Allegri spoke, who started his second spell at AC Milan his frustrations.
“If we look at the implementation, we had eight shots on goal, but the difference is made with the determination with which you attack and defend,” Allegri told Dazn Italia.
“We have admitted two avoidable goals that we could have prevented with a little more concentration. Fortunately it is only the first round and we have time to work on these details.”
“We all have to prepare ourselves in a certain way, because we cannot continue to admit such soft goals.”