The Rossoneri go into Sunday's game against Serie A leaders Napoli buoyed by their 2-0 triumph over the Scottish champions in their Group H opener on Wednesday and Allegri has called for more of the same.
"Against Celtic we suffered and we did some good things and some bad things but at least we know how to suffer," Allegri said.
"My team has yet to grow not only from a physical standpoint but from a mental one. We need to increase our self-esteem and the only way to do that is to win games."
Milan have made a slow start to the new domestic season, winning just one of their opening three league games and they were held to a 2-2 draw at Torino last weekend.
However, they have been hit with injuries early on. Ignazio Abate, Mattia De Sciglio, Matias Silvestre, Daniele Bonera, Kaka, Riccardo Montolivo and Giampaolo Pazzini are all ruled out through injury.
Allegri knows his team face a stern test against a Napoli side that has the best attack in Italy's top flight.
Napoli have found the net nine times in three games.
"We know our limits," Allegri said.
"Napoli have a great team and (Napoli coach) Rafa Benitez is doing a great job.
"We know it is going to be a difficult game but we are determined to do well against them."
Rivals Napoli also enjoyed success in the Champions League this week.
They beat Group F rivals Borussia Dortmund 2-1 at the Stadio San Paolo to remain perfect this season.
Under Benitez, who replaced Walter Mazzarri this summer, Napoli are showing no signs of slowing down and currently lead the standings alongside Roma with nine points.
Last weekend's 2-0 triumph over Atalanta marked their third straight win in Serie A.
"Our aim is to go as far as possible in every competition," Napoli goalkeeper Jose Reina said.
"Our start to the Champions League has been wonderful. Now we must focus on the game against Milan."