The England soccer team has developed a Plan B for Euro glory.

But while powerhouses like Germany, Italy and Portugal all went home, England stayed in. They overcame a one-goal deficit to beat Switzerland on penalties on Saturday to reach the semi-finals for the third time in the last four major tournaments.

What the win made clear was that England's Plan A might fail. But with Plans B, C and D, this team has become one of the most tenacious tournament teams in world football. England scored with almost the last kick of regulation to tie the quarterfinal against Slovakia before winning in extra time, and then needed Bukayo Saka's 80th-minute goal to tie the game against Switzerland on Saturday and send the game into 30 minutes of extra time.

“This is not a normal football game,” said manager Gareth Southgate as he prepares to face the Netherlands in Wednesday's semi-final. “It's a national tournament with a lot of pressure and some really young players in it. Our team has been under a lot of pressure from the start but they've done very well.”

It's hard to argue with the results, but everything else has been a source of endless angst back home. Southgate has been criticised for everything from his starting XI to his tactics to his outfits. The public seemed to like him best when he wore a vest at the 2018 World Cup. Above all, fans are frustrated with England's boring football. England have scored just five goals in five games and averaged 3.2 shots per game, fewer than any team that made it to the knockout stages.

“We have an ideal vision for ourselves,” Southgate said, “and we've had to find a way to win despite all the obstacles we've had to overcome.”

At times, finding those solutions looked downright chaotic – against Slovakia, Southgate made two attacking substitutions after the 83rd minute, bringing on forward Ivan Toni with just seconds left – but it paid off, as Toni scored a vital header to win the game.

Similarly, England finished extra time against Switzerland with a rag-tag midfield made up of attack-minded players, including several who were on the pitch solely for the ability to take penalties. England played quite differently to their original tactics, but planning for the unexpected and delivering is what England is all about. Southgate has seen many English teams, including some he played for, start well only to be eliminated in the first knockout round.

“We didn't have the knowledge, we weren't familiar with the tournament,” Southgate said of previous generations of England players. “This group is different… Generally we have shown the tenacity of the teams that win tournaments, the France and the Italy. It's not just pure football. We've shown a bit more street smarts.”

Nowhere is England's attention to backup planning more evident than in how they've approached penalty shootouts. For decades, penalty shootouts have been dismissed as the product of luck that has always worked against England. Anyone who followed England in the 1990s and 2000s knows the list of disastrous results: Germany (1990 World Cup), Germany again (Euro 1996), Argentina (1998 World Cup), Portugal (Euro 2004), Italy (Euro 2012).

But under Southgate, who missed the winning penalty in 1996, the tournament's penalty shootout was simply another aspect of football, studied, analysed and reconstructed. England started to call up more players who regularly took penalties for their clubs. Players learned to slow down their approach to the ball. And they gave goalkeeper Jordan Pickford reams of data on opposing penalty takers, sticking it in the form of a cheat sheet on Pickford's water bottle to refer to during the shootout.

Suddenly, a country where penalty shootouts were a national neurosis was winning them. England scored all five of their shots on Saturday, giving Southgate his third shootout victory in four attempts. The only exception happened to be the Euro 2020 final against Italy.

“I think we have a good process,” Southgate said. “We've played four times, won three times and, of course, we were totally blamed for the one we lost but that's always the case.

Despite this defeat, Southgate's record makes him arguably England's greatest manager of all time, winning around 60% of his games as manager and raising the bar for tournament semi-finals to the bare minimum. It's easy to forget that England went 12 years without winning a knockout match during his tenure, until their first World Cup appearance in 2018.

During that time, England missed out on some tournaments but perhaps would have liked to have played in others, such as Euro 2016, where they suffered a humiliating defeat to Iceland in the last 16. England, the inventors of football, were fumbling around in major tournaments and finding new ways to embarrass themselves.

Under Southgate, England have won eight Euros and World Cup knockout stages, compared with just six in the previous 40 years when the tournaments featured fewer teams.

“I took this job to improve English football,” Southgate said. “I wanted to restore credibility on the world stage.”

In that respect Southgate has been successful. England are now far from the laughing stock they were after their defeat to Iceland. But as expectations rise the pressure on Southgate is mounting and whatever happens now it seems unlikely he will stay on as manager beyond the Euros. Which is why Southgate savored the moment on Saturday and celebrated with his players on the pitch.

“Sometimes I think there must be something fun about this job,” he said.

Write to Joshua Robinson at Joshua.Robinson@wsj.com.