Allegri rues Milan performance
Two late goals from midfielder Claudio Marchisio handed the Turin giants a deserved win, and condemned the champions to their second Serie A loss of the season.
"We were not up to the standard required," Allegri said. "We made too many mistakes. Juve were better than us.
"This was our worst performance of the season so far, and this defeat should be a lesson that in order to win, we have to play at a certain level."
Milan have endured a slow start to the campaign and have collected just five points from five games. However, Allegri is not pushing the panic button yet.
"I don't think there is a psychological problem," he said. "We weren't as sharp as we could have been against Juve.
"We didn't play well in defence or in attack. But there is nothing to worry about."
Milan, who head into the international break six points behind joint leaders Juventus and Udinese, welcome Palermo to the San Siro on 15 October.