The former Juventus and AC Milan boss believes his side should be angry after losing focus and determination at both ends of the pitch to drop two points.
Speaking to DAZN Italia after the match, Allegri assessed the performance as solid in parts but ultimately insufficient.
"I thought it was a good technical performance; there were the two incidents off the crossbar, but at times we need to be more determined in front of goal at both ends," he said.
"So much depends on the determination with which you attack the ball, both going forward and when defending in your own area."
The coach was particularly pointed about the mentality required after dropping points at home.
"We might be growing, but if we don't leave the field angry today, then it means we've still got a very long way to go," Allegri added.
"Having lots of possession isn't enough and we ought to be creating and converting more scoring opportunities.
"We train to improve everything, but what we absolutely must do is show more focus, because with a minute to go we lost the ball cheaply and were caught unbalanced. We need in those situations to at least bring home the result."
Looking at the broader picture in what he described as a competitive season, Allegri struck a measured tone while stressing the need for improvement.
"I think there are many strong teams this season, so it's a very difficult league. We must try to recover the players with certain characteristics, stay calm and do more," he said.
"We need to leave here angry today, because we could've done better in various areas, but we've got to be angry at this result; otherwise, it's worrying.
"We need to work on our limitations, because if we only assume we're better than the opponents, we'll pay the price. That is something we absolutely must improve."