"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.
"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."
