Irate supporters blocked exits and demanded a meeting with the team following their dismal 1-1 draw with 10-man Genoa on Saturday
The Rossoneri went up 1-0 after only four minutes but were pegged back immediately by the dogged visitors, who survived a subsequent siege to their goal despite being a man down for the second period.
Supporters watched on as Allegri's men ran out of ideas to beat an inspired Mattia Perin, whose heroics between the Genoa posts included saving Mario Balotelli's 36th-minute penalty.
Irate fans from San Siro's Curva Sud section blockaded the stadium exits after full-time, demanding a personal audience with the team and Allegri over the side's state of affairs.
"These players don't know what sacrifice is. They have so much money, while people like us work hard for them," a spokesperson told reporters.
The Ultras had earlier unfurled a banner proclaiming "Let's meet by the exit you unworthy lot", prompting police to form a cordorn around the players' garage.
Balotelli voluntered to pacify them but was stopped by security personnel, so Kaka and goalkeeper Christian Abbiati went to meet a body of 300 fans in the striker's stead.
The pair spent three minutes in heated dialogue with the group outside the stadium grounds before being allowed back to the dressing rooms to widespread applause, with the situation apparently resolved.
"Things are not going well this year, but that happens to everybody. There's no need to overdramatize," club CEO Adriano Galliani appealed.
"I won't speak about the Curva Sud tonight. You need to be patient and stay calm. We will get back on track."
"I am eager to bring the team back to a good position in Serie A, continue in the Champions League and also the Coppa Italia," coach Allegri pleaded. "I have not run out of ideas yet, or I would have left.
"It's only natural there is bitterness that we are without a win in a month. In my three years at Milan we always did well, but things aren't going our way this season.
"Balotelli must stay calm, as penalties can be missed. He's scored so many penalties before, but unfortunately the last two he took were saved."
Milan are now without a win in their last seven matches in all competitions and sit 18 points behind leaders Roma in Serie A.