The Rossoneri failed to impress against Inter and Juventus at the Trofeo TIM, but then beat Schalke 1-0 before recording a 1-0 win over Chelsea at the weekend, and Allegri is happy with his side's recent improvement.
"It was a good test and the players performed well. We played well defensively, we were well organised and we worked well up front, creating a good number of chances," Allegri was quoted as saying on the club's official website.
"It was good that we finished the match without suffering any injuries; so, we can say that we are working well in view of the league campaign. We have made good progress.
"In the first half we came close to scoring, choosing the wrong final pass at times. We didn’t run many risks at the back, only when there were scrambles in and around the penalty area. I am satisfied."
Milan start the new Serie A season at home to Sampdoria on August 25.