The defending Serie A champions began the match with promise but it was a fumble from shot-stopper Abbiati that gave visiting striker Antonio Di Natale an easy tap in for the opening goal.
A Stephan El Shaarawy equaliser in the 64th minute tied the score at 1-1, rescuing the San Siro outfit from what would have been a disappointing loss.
Abbiati told the club's official website: “Udinese's goal came from my mistake, I am sorry for the team. I also apologised to the fans after the match.
“In the second half, the team found the equaliser and came close to victory.
“In a moment like this there will be some unfortunate moments, but we had a great performance against the team that, in my opinion, is the fittest in the league.”
The draw sees AC Milan take just two points from their opening three games, leaving them in 15th position.