Southgate said he heard it, too.
"In this day and age, you are ashamed that it can happen," the England manager said. "The joy of a young player, who has had such a fantastic game, to be overshadowed by that is hugely disappointing."
Southgate said the English Football Association would be making an official complaint to Uefa.
"It's not acceptable," he said.
In one unsavoury incident, Sterling cupped his ears after scoring England's fifth goal and Montenegro fans responded with jeers before an object — reportedly a lighter — was thrown on to the field. It was picked up by Hudson-Odoi.
Sterling has been vocal about racism in recent weeks, highlighting what he perceived to be negative press coverage towards black players and saying there needed to be more done to combat it.
After the game in Montenegro, he posted a picture on Twitter of him cupping his ears.
"Best way to silence the haters (and yeah I mean racists)," Sterling wrote next to an emoji of a monkey and a hashtag saying "get some education".
Hudson-Odoi said Sterling and Rose had spoken to him about racism in football.
"They've said, 'Listen, in football you're always going to get things like that. People are always going to be rude to you, say stuff you don't want to hear. It's part of football,"' Hudson-Odoi said. "You've just got to block it out of your head, keep going. But at the same time, that should never happen in football. Everyone should be enjoying the game."
The Fare network, Uefa's anti-discrimination monitoring partner, had a delegate at the game and has been gathering evidence.
"When you hear stuff coming like that from the fans, it's not right," Hudson-Odoi said. "It's unacceptable. Hopefully Uefa deal with it properly."
Ross Barkley scored twice — one of the goals a glancing finish off Hudson-Odoi's shot — while Michael Keane and Harry Kane added the others for England, who started their qualifying campaign with a 5-0 win over the Czech Republic on Saturday. It is the first time England have scored five or more goals in successive games since 1984.
Montenegro took the lead against the run of play in the 18th minute, Marko Vesovic taking advantage of sloppy defending to curl home a finish.
In the group's other game, Kosovo marked their first Euro qualifier since becoming a member of Uefa in 2016 by drawing with Bulgaria 1-1 at home.
- AP