Despite finding his team 4-0 down on Saturday night with two goals conceded in each half, Allegri saw positive aspects of Milan's play, though he admitted that their defending left a lot to be desired.

"We definitely have to improve our defending from set plays, while we were too disorganised and hasty in attack," Allegri told reporters.

"After the break we were trying to get back into the game and conceded another two goals against the run of play, while the Roma goalkeeper had to perform quite a few tough saves."

Allegri offered credit to his opposition, who currently occupy the sixth spot in the Serie A table, only a single place above Milan.

"I think the Giallorossi are one of the best teams in Italy and I am amazed we are so close to them in the table," Allegri said.

The coach insisted his side has more work ahead and suggested the loss of several high-profile players during the summer had lead to increased and undue cynicism among fans and pundits.

"People think we go from the stars to the gutter in a millisecond," Allegri said.

"Don't forget we went into this game with four wins and a draw in six games. Of course we can do better and the club will certainly do something in the transfer window, but it's not as if we were challenging for second place before this defeat.

"Sometimes games go against you and we contributed to the result, but at the end of the day I think Milan played a good game.

"Roma deserve credit for making the most of the space we left and we know how dangerous Zdenek Zeman's teams can be in these situations."

Roma coach Zdenek Zeman was thrilled with the result, but believes his side still have plenty to work on.

"We need to think one game at a time. We defeated Milan tonight after a positive series of good victories for them. I'm happy with the guys' performance, we need to look ahead and prepare well for the rest of the games we have," he said.

"I think we can improve both in defence and in attack. The defence department should follow the rest of the team, the midfield should give directions to follow and the strikers should sacrifice themselves a bit. Totti did a great job for the defence tonight and that's very important for us. In attack we're still making too many mistakes when we have easy chances to score. We need to improve that aspect."