England lost on penalties at Wembley Stadium on Sunday night, with Marcus Rashford, Jadon Sancho and Bukayo Saka missing their spot-kicks.
The trio were subjected to racist abuse on social media after the game.
“It’s just not what we stand for,” Southgate told a news conference on Monday morning.
“We have been a beacon of light in bringing people together, in people being able to relate to the national team, and the national team stands for everybody – and so that togetherness has to continue.
“We have shown the power our country has when it does come together and has that energy and positivity together.
“Everybody has to remember when they support they team that they also represent England and should represent what we stand for.
“We can’t control [the behaviour of a minority of supporters]. We can only set the example that we believe we should and represent the country in the way that we feel.
“I think the players [...] have had a positive effect on lots of areas of society, but we can’t affect everything. Other people have responsibilities in those areas and we’ve all got to work collectively to constantly improve those things.”
Meanwhile, the Duke of Cambridge tweeted: “I am sickened by the racist abuse at England players after last night’s match.
“It is totally unacceptable that players have to endure this abhorrent behaviour.
“It must stop now and all those involved should be held accountable.”
Earlier on Monday, Prime Minister Boris Johnson wrote on Twitter: “This England team deserve to be lauded as heroes, not racially abused on social media.
“Those responsible for this appalling abuse should be ashamed of themselves.”
A spokesperson for the FA said: “We could not be clearer that anyone behind such disgusting behaviour is not welcome in following the team. We will do all we can to support the players affected while urging the toughest punishments possible for anyone responsible.
“We will continue to do everything we can to stamp discrimination out of the game, but we implore government to act quickly and bring in the appropriate legislation so this abuse has real life consequences.”
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