Shortly after Milan had scored to take a 1-0 lead and Milan were about to take a goal kick, Maignan signaled to his teammates to stop, walked to the referee and then to the sideline.
"They made monkey noises and it's not the first time it's happened to me," the 28-year-old Maignan told DAZN after Saturday's match.
Teammates attempted to console Maignan on the sideline but then the France international took his gloves off and walked down the tunnel leading off the field.
Maignan had also told the referee about monkey chants earlier during the match, prompting an announcement in the stadium asking fans to stop.
The Serie A game resumed about five minutes after the suspension when Maignan and teammates came back out. Milan won 3-2 and moved within four points of second-place Juventus, who visis Lecce on Sunday.
"I was upset to have to go into the changing room like that but I had everyone's support," Maignan said. "We talked and then we made the decision to go back out onto the field and make the appropriate response - to win this game."
"I really didn't want to play anymore. … But we're a family and I couldn't leave my teammates like that."
Udinese will likely be hit with a partial stadium closure for their next home match.
"There is absolutely no place in our game for racism: we are appalled, " Milan said on X, formerly Twitter. "We are with you, Mike."
There have been numerous racist incidents in Italian and European soccer for years, with cases in Italy aimed at Kevin-Prince Boateng, Mario Balotelli and Romelu Lukaku among others.
Luke Jovic and Noah Okafor scored late as Milan came back from a goal down to secure the victory.
Jovic knocked in the rebound of a shot from Olivier Giroud that hit the crossbar in the 83rd and then Giroud headed down a corner toward Okafor for the winner in stoppage time.
Lazar Samardzic and Florian Thauvin had put Udinese ahead after an early Milan goal from Ruben Loftus-Cheek.
In Rome, there were protests outside the stadium following the firing of Jose Mourinho.
Inside the Stadio Olimpico, new Roma coach Daniele De Rossi - the former Giallorossi captain - was greeted by loud cheers. Yet there were also numerous banners saluting Mourinho.
It was a unique atmosphere for De Rossi's debut as Roma quickly shed Mourinho's defensive style for a more offensive approach in a 2-1 win over relegation-threatened Hellas Verona.
Romelu Lukaku and Roma captain Lorenzo Pellegrini scored six minutes apart during the first half, with both goals set up by Stephan El Shaarawy.
Michael Folorunsho pulled one back for Verona with a long, swerving shot midway through the second half after teammate Milan Djuric missed a penalty kick, setting up a nervy finale for Roma.
The victory moved Roma within two points of the top four and the Champions League zone - the objective De Rossi was charged with when he signed a six-month contract.
