Raheem Sterling celebrated the fifth goal on Monday by cupping his ears towards the home fans as a sign of defiance, while Callum Hudson-Odoi said he and Danny Rose heard monkey noises aimed at them.
"Sanctions are worthless if there is nothing alongside that to help educate people," said England coach Southgate.
"My kids don't think for one minute about where people are born, what language they speak, what colour they are.
"There's an innocence about young people that is only influenced by older people. So we have to make sure that the education is right for everybody.
"In our country, (it's) the same. I've said this before, I'm not sitting here just criticising what's happened tonight because in our country we have the same issue. We're not free of it."
Sterling's fifth capped a fine performance in which Ross Barkley struck twice and Michael Keane and Harry Kane also scored as England overcame the hosts' early opener to seal an impressive victory.
England coach Southgate said the racist chanting had marred the win, which leaves England in charge of Group A following two big wins.
"What we have to do is make sure our players feel supported," Southgate told BBC Radio. "They know that the dressing room is there and we're there as a group of staff for them. We'll report the incidents through the correct channels. It's clear that so many people have heard it."
"We have to make sure that in our country we continue to make the strides and that we've got to trust the authorities to take the right action here."
UEFA regulations state that if supporters engage in racist behaviour then "the member association or club responsible is punished with a minimum of a partial stadium closure".
Article 14 also states that additional sanctions can be imposed depending on the situation, while "disciplinary measures may be combined with specific directives aimed at tackling such conduct".
UEFA said on Tuesday that disciplinary proceedings had been opened and charges against Montenegro including racist behaviour and crowd disturbances.
The case will be dealt with on May 16.
Sterling tweeted a picture of his goal celebration, saying: "Best way to silence the haters (and yeah I mean racists)".
"It's 2019 now, I keep saying it. It's a shame to see this going on and we can only bring awareness to the situation," he told Sky Sports.
"It's now time for the people in charge to put a real stamp on it because you can fine someone, but what is that going to do?"
Football Against Racism in Europe (FARE) praised the England players for their reaction in a match it said it had designated as "high risk" prior to it taking place, and called for heavy sanctions.
"We commend the reaction of the England players involved. No human being should have to face abuse and vilification for their race or identity, something that many Montenegrins will understand from the divisive and bloody recent history of the Balkans," FARE said in a statement.
"We hope that UEFA will act decisively, the sanctions that could be applied for an offence of this kind range from a partial stadium closure to full stadium closure."
Kick It Out campaigner Troy Townsend said the incidents showed UEFA had "failed the game" in its handling of racist abuse
Southgate has called for education to stamp out racism in football as European governing body UEFA opened disciplinary proceedings against Montenegro.