The visitors took the lead through Andrea Poli in the first half but were beaten by two Luca Toni headers either side of the break.
Milan finished third last season and were favoured to win the match but failed to hit their stride, leaving Allegri to rue three points dropped.
"We read the match wrong after taking the lead," Allegri said.
"We played well for the first 15 minutes but after the goal we were wrong technically and temperamentally and we stopped being aggressive.
"If we want to get results we must have a different spirit."
The 46-year-old went on to praise veteran striker Toni, who marked Verona's first Serie A game in 11 years with a classy double.
"I offer congratulations to Toni," he said.
"He fought like a lion between the two central defenders and he serves as an example to our kids."
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