England players were also pelted with cups thrown from the hostile Puskas Arena crowd, with many Hungary fans also booing Southgate's team when they took the knee before kick-off to show support for equality.
Southgate was impressed with the cool response from Manchester City forward Sterling, who scored England's opening goal against the Hungarians, and Borussia Dortmund midfielder Bellingham.
"Unfortunately, I don't know how many camps in the last four years but I seem to have been talking about this subject almost every time we have been together," Southgate told reporters on Saturday.
"I can only reiterate that our players are incredibly mature in the way they deal with it. I think they feel supported by their team-mates, which is very important to them.
"I think their team-mates recognise how challenging it must be for our black players and how disappointing it is in the modern world that we continue to have to answer these questions because of the incidents that happen.
"But we can only keep taking the stance that we have done and hope that we continue to send the right messages, not only to people in football but across society, and that everybody keeps progressing.
"We know it's going to take time and we know that feels very slow for everybody, but we have to keep fighting that battle."
England defender Conor Coady said the venom from the fans in Hungary had drawn the squad even closer together.
"It's something we don't want to speak about, but it's happening and I think with the way the boys deal with these situations, because we're such a close-knit group, it's something where once it happens to one person, it happens to us all," Coady said.
"We'll carry on doing what we're doing, carry on trying to take a stance and in terms of not just in football but in society we want to try to help, and help change.
"I think it's important we all stick together, which is something we'll always do and the two boys, they are absolutely fantastic people, but we're a team that will always get around them as much as we possibly can."
With England sitting top of Group I and a crucial clash with second-placed Poland looming on Wednesday, Southgate plans to make several changes for Sunday's qualifier against Andorra at Wembley.
Wolves centre-back Coady is set to feature, while there could be an England debut for Leeds striker Patrick Bamford.
"It's a special moment for him and for his family. You always think about people that have helped you on that journey," Southgate said of Bamford.
"We want him to do what he does every week with his club. He has settled in really well with the group, he hasn't had a huge amount of time to be able to train with us yet but I think he's got a really clear picture of how we like to play.
"We just want him to go and enjoy his football. It's an opportunity and he shouldn't put any more pressure on himself than that. We know what he's capable of and we're looking forward to seeing him play."