Federico Baschirotto, 28, scored the first goal for Cremonese after a mistake by Strahinja Pavlovic allowed him to score from close range.
Federico Bonazzoli equalized for Cremonese just after the hour mark with an impressive overhead-kick, beating Mike Maignan.
Allegri shared his thoughts with the press after the game, reflecting on the team's performance.
"If we look at the game, we had eight shots on goal, but the difference is made with the determination with which we attack and defend," Allegri said.
"We conceded two avoidable goals that we could've prevented with more concentration. Fortunately, it's 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 easily scored goals."