Maignan alerted the referee before leaving the pitch after the half-hour mark, followed by his team-mates.
The incident took place shortly after Milan midfielder Ruben Loftus-Cheek had given the visitors the lead with a low first-time shot inside the post.
The Serie A match was restarted after about 10 minutes.
"I said we cannot play football like this. It is not the first time it has happened this way," Maignan told Sky Sports Italy.
"They must hand out very strong sanctions, because talking no longer does anything.
"We have to say that what they are doing is wrong. It is not the whole crowd, most fans want to cheer on their team and jeer you. That's normal, but not this."
He added: "What happened in the first half is that at the first goal-kick I heard monkey noises, I said nothing.
"The second time I went to get the ball, I heard it again. I told the fourth official and my bench what happened.
"I was angry, because this isn't the first time it has happened. I did not want to play, but we are a family and I could not leave my team-mates like that."
AC Milan posted on X: "There is absolutely no place in our game for racism: we are appalled. We are with you, Mike."
League leaders Inter Milan replied: "We are brothers of the world, against all forms of discrimination. By your side."
Milan match-winner Noah Okafor told DAZN: "These things have nothing to do with football and they hurt, there is no such thing as black or white. We are all the same."
Former England and Arsenal striker Ian Wright also wrote on X: "Solidarity from the rest of the team. Keep walking off!! We did 'playing through it' and nothing has changed. Points deductions needed, the fines are pointless."
Serie A also "condemned any form of racism".
France keeper Maignan, 28, was beaten three minutes later by a Lazar Samardzic shot before Florian Thauvin put Udinese in front in the 62nd minute.
But Milan snatched victory when substitute Luka Jovic netted a rebound with seven minutes left and Okafor made it 3-2 in stoppage time from close range.
Milan are third in Serie A, six points behind leaders Inter Milan and four adrift of second-placed Juventus, who both have a game in hand.