The head coach watched as his side threw an early advantage to trail 2-1 to the Serie A strugglers with 10 minutes remaining at the Stadio Armando Picchi.
Mario Balotelli – who opened the scoring for Milan – rescued them a point with a fantastic free-kick six minutes from time, but Allegri condemned his side for a lack of intensity.
"I compliment Livorno because they put in a very good performance," he told Sky Sport Italia.
"We showed a lack of determination and it is difficult to win a match like that. We conceded the second goal from a throw-in and that is not good enough."
"Both M’Baye Niang and Mattia De Sciglio did well down the right flank, but we paid a heavy price for the mistakes in defence.
"We've got to understand that if we don't match our opponents in terms of determination and intensity, then our quality won't make the difference."
The 46-year-old did have praise for Balotelli after the 23-year-old struck his fourth goal in three games for Milan.
"Balotelli had a great game and when he plays like that he is difficult to mark," Allegri added.
"He should've had better support, not so much in terms of passes as in attitude."
