Southgate was the mastermind of England’s first win against Germany in the knockout stages of a major tournament since 1966, with Raheem Sterling and Harry Kane scoring the goals.
His team selection was roundly criticised before the game on social media — particularly his decision to leave out Jack Grealish — but Maguire insists he was proved right.
“Credit to the coaches and Gareth,” said the Manchester United defender. “I thought the game plan worked brilliantly, we executed it really well.
“You can’t please everybody in terms of picking a starting XI, especially when it is your country. Gareth picks the one he thinks is the best one to win a football match and I am sure he will continue to do that without listening to what everyone else thinks.
“Everyone has an opinion and that’s football. Leadership is a big part of where we are now.”
Southgate savoured his landmark win, but had a warning for his players ahead of Saturday’s quarter-final.
“It didn’t really need me to say it, but when we got in the dressing room we were talking about Saturday already, because today has been an immense performance but at a cost, emotionally and physically,” he said.
“We’ve got to make sure we recover well and mentally we’re at the right space. It’s a dangerous moment for us.
“We’ll have that warmth of success and that feeling around the country that we’ve only got to turn up to win the thing. We know it’s going to be an immense challenge from here on. The players know that.
“They’ve been to the latter stages before — they know how difficult it has been. Their feet are on the ground, they should feel confident in the way they’ve played, the manner of their performance. But we came here with an intention and we’ve not achieved that yet.”
Southgate added: “We can look back on a day like today in the future, but I want to get Saturday right and get my focus on that game.
“Keep calm. I know what we set out to achieve and today, of course, is a big result. England have never managed to beat Germany in a knockout game since the  final here. I think we haven’t put together a semi-final (the 2018 World Cup) followed by a quarter-final since 1966 either, so these players keep writing history and they’ve got the chance again.
“We’ve only ever been to one European Championship semi-final, so again they’ve got the chance to do something really special and we’ve got to make sure we give ourselves the best chance of doing that.”
Captain Kane echoed Southgate’s words, saying: “To hear Wembley like this was a moment none of us will ever forget. It’s the perfect afternoon. We will enjoy this one but we have another big one on Saturday now.
“Any team will be looking at us and know that we’re dangerous. We need to get it right on Saturday, we can’t switch off. We want to go all the way, so need to keep it up.
“There isn’t anything bigger than this when the expectation is on and we delivered, so we should be proud, but we can’t stop here.
“We have a vision of where we want to go and we can’t stop now. Hopefully we’ll be back here in the semi-finals and the final.”