Allegri given vote of confidence after defeat
"We'll keep going with this coach. He enjoys the complete faith of both the club president [Silvio Berlusconi] and myself," chief executive Adriano Galliano told Milan's website.
"Milan played better, I've seen a team that is improving. Now, I'm hoping for a win at the San Siro on Wednesday even though it seems to be cursed for both us and Inter," he added.
Milan, held 0-0 at home by Anderlecht in the Champions League last Wednesday, have yet to win at the San Siro in three attempts in all competitions this season.
Rivals Inter, who share the stadium with them, are five matches without a win at home, two in Serie A and three in the Europa League.
Allegri, whose side host Cagliari on Wednesday, led Milan to the Serie A title in his first season in charge in 2010/11 and to second place last season.
This season, they have lost three of their first four league games after selling several top players, including Zlatan Ibrahimovic, Antonio Cassano and Thiago Silva, to cut costs.
"It seems like lady luck has turned her back on us this year, but we're only at round four. It's early," added Galliani.
Allegri, whose side had defender Cristian Zapata and midfielder Kevin-Prince Boateng sent off, said: "I don't think the players deserved to lose.
"We've conceded too many times from dead balls and we need to improve on that. We should have managed the ball better after a good opening. Apart from Juventus, who are stringing together a load of wins, the other teams are not far off.
"It's a comforting performance, a game played with courage that will give us faith.
"I'm in touch with Adriano Galliani every day and I'm in touch quite often with president Berlusconi during the week and so there are no problems there. Now we need to break the San Siro taboo on Wednesday."