The Rossoneri squandered the chance to leapfrog Roma in the Serie A table following the capital side's defeat at Catania, and while their coach was satisfied to have kept a clean sheet, he conceded work has to be done in the final third of the pitch.
"We have thrown two points away," Allegri told Sky Sport Italia. "Before [the match] we had a team that had almost beaten Juventus but played well.
"We came [to Genoa] on the back of 120 minutes in the Coppa Italia. We had many chances and missed several openings, but it is important not to have conceded a goal. The back four worked pretty well tonight.
"I saw a good game, but it is clear that in the final third, things must pick up. Also, we still have to eliminate individual errors."
Despite the disappointing nature of Milan's performance, Allegri offered words of encouragement to teenage French striker M'Baye Niang, who made his league bow and completed 90 minutes in Liguria.
"He had a good game," he added. "It is not easy for such a young lad, even [Mattia] De Sciglio and Bojan [Krkic] made a little effort, but he was alone at the back and it was not easy.
"These games are usually decided by a bit of magic. I think Milan should be seen in a different light than in previous years, given the situation we find ourselves in."