The Serie A new-boys were good value for the three points, with Allegri asked afterwards if his squad was short of quality and depth.
He began: "We had to expect this match: Cremonese, a tough team that doesn't let you play, was our first game of the season. We just had to raise our threat perception tonight, as we will in the next 37 games.
"Otherwise, we're a team that can lose to anyone. We need to raise our level of aggression, both offensively and defensively, and be aware of the danger. We knew Cremonese were strong on crosses, and conceding two goals like this is just a matter of threat perception."
Asked if the squad is good enough, Allegri continued: "I only know one thing: this team has excellent players. We didn't lose tonight because (Rafael) Leao or anyone else were missing... We lost because you have to be aware of the danger: when you don't have the ball and you don't sense the danger, you can concede at any moment.
"You have to raise your focus and your perception. We created chances, but then you concede with the first cross, and that's no good. It doesn't exist. The Italian league is tough, you have to win even the dirty games, especially the ones at the start of the season, and tonight we were wrong about that. It's not a tactical or technical issue, but that you have to make a leap forward, a switch... Maybe you weren't winning tonight, but you certainly weren't losing."
On the final week of the market, he added: "The club is evaluating the situation and has done a good job this summer. Right now, we need to significantly increase our focus, even if that means making one less play in attack. The first thing to do is reduce the number of goals we've conceded, otherwise it's impossible. We can't leave the pitch having conceded two goals when we were in numerical superiority."