A first-half strike from Robinho and a 25-yard curling effort from Nigel De Jong put the visitors 2-0 ahead in the Guinness International Champions Cup encounter, and they held on for victory despite Dani Parejo's goal for Miroslav Djukic's men.
Speaking to SportMediaset, Allegri conceded that he had concerns over his side's fading fitness levels late in the game but assured fans that improvement was inevitable.
"It was a very intense match," the 45-year-old said. "I wanted to see if the team was in the right physical and psychological condition. We need to improve, as at the end we were struggling to stay on our feet, but for the moment I am satisfied.
"It was important to have a strong performance, especially in defense. Some players need a little more time on the field, but we always play to win, as victories help you to work better.
"We’re on the right track, though we have to do better in the final third. I am glad Nigel De Jong got his goal, as he deserved it."
The Rossoneri's next fixture is against Manchester City on Wednesday with Allegri hopeful of including Mario Balotelli against his former club for his first preseason appearance.