Federico Baschirotto, 28, opened the scoring for the visitors after Strahinja Pavlovic’s mistake allowed the Cremonese to nod home from close range.
Federico Bonazzoli restored their lead just after the hour mark with an early goal of the season contender, smashing an acrobatic overhead-kick past Mike Maignan.
Speaking to the press after the game, Allegri, who started his second spell at AC Milan voiced his frustrations.
“If we look at the performance, 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 conceded two avoidable goals that we could’ve prevented with a little more concentration. Fortunately, it is only the first round and we have time to work on these details.”
“We must all prepare in a certain way, because we cannot keep conceding such soft goals.”