Carlos Tevez had given the Derby d'Italia hosts a fifth-minute lead but Mauro Icardi levelled for Inter in the second half.
Inter had the better of the chances after the break, and also had Mateo Kovacic sent off late on.
Allegri said his league leaders were at their best for the opening 50 minutes but rued their inability to maintain their level.
"It's a shame we weren't able to win our first match of 2015," he told a media conference.
"We knew how difficult the game would be but we put in one of our best performances of the season in the first 45 minutes, creating a great deal and not allowing them a shot on goal.
"However, we lost our way a bit before Inter's goal, something that shouldn't have happened."
Allegri felt a lack of movement was costly for his side during the second half and panic set in when errors came.
"When we make mistakes we lose our calm," he said.
"We were too static when moving the ball around in the second half, while in the first half the midfielders made themselves difficult to pick up.
"We must improve and manage to maintain our level of display for the entire game, not just 50 minutes."
