England’s soccer team has landed on a plan for Euro glory. It’s called plan B.

And yet, as heavyweights such as Germany, Italy, and Portugal all went home, England has stuck around. Now, after coming back from a goal down to knock off Switzerland in a penalty shootout on Saturday, the Three Lions are semifinal-bound for the third time in their past four major tournaments.

What the victory revealed was that England’s Plan A might be a dud. But Plans B, C, D and onward have turned this team into one of the most resilient tournament sides in world soccer. The Three Lions scored with practically the last kick of regulation to tie their round-of-16 game against Slovakia before winning in extra time. And on Saturday, they needed an 80th minute goal from Bukayo Saka to draw level against Switzerland and force an extra 30 minutes.

“These are not normal football matches," said manager Gareth Southgate, who must now prepare his team to face the Netherlands in Wednesday’s semifinal. “These are national events with huge pressure, with really young men in the middle of it. Our team has been under enormous pressure from the start, and they’re doing so well."

The results are hard to argue with. Everything else, however, is a matter of unending consternation back home. Southgate has been criticized for everything from his starting lineups, to his tactics, to his sartorial choices—the nation seemed to like him best when he was sporting waistcoats at the 2018 World Cup. Above all, fans are frustrated with England’s brand of stultifying soccer. It has scored just five times in five matches and has averaged 3.2 shots on goal per game, the fewest of any team to reach the knockout rounds.

“There’s what we ideally want to be," Southgate said, “and there’s how we’ve needed to find ways to win with all the obstacles we’ve had."

At times, finding those solutions has looked downright chaotic. Against Slovakia, Southgate made two attacking substitutions after the 83rd minute, including throwing on forward Ivan Toney with mere seconds to play. But the move paid off as Toney played a vital header to assist the winning goal.

Similarly, against Switzerland, England finished extra time with a patchwork midfield made up of attack-minded players, including a couple who were on the pitch strictly for their penalty-taking ability. The Three Lions’ play looked nothing like their initial tactics, but contingency planning and grinding out results is what this version of England is about. Southgate had seen too many England sides—including some he played in—start matches well only to be caught out and eliminated in the first knockout round.

“We weren’t savvy, we weren’t tournament-wise," Southgate said of England’s previous generations. “This group are different…In general we’ve shown the resilience that teams that win tournaments have—France, Italy. It’s not all pure football. We’re showing a little bit more of that streetwise nature."

Nowhere is England’s attention to its backup plans clearer than in its approach to penalty shootouts. For decades, those exercises had been written off as a roll of the dice that somehow always broke against England. The list of heartbreaks was known by heart to anyone who had followed the Three Lions through the 1990s and 2000s: Germany (1990 World Cup), Germany again (Euro 1996), Argentina (1998 World Cup), Portugal (Euro 2004), and Italy (Euro 2012).

But under Southgate, who had himself missed the deciding spot-kick in 1996, tournament shootouts were just one more aspect of soccer to be studied, broken down, and built back up. England made sure to call up more players who were regular penalty-takers for their clubs. It taught them to slow down their approach to the ball. And it armed goalkeeper Jordan Pickford with a mound of data on opposition penalty takers—which it then glued to his water bottle in the form of a cheat sheet for Pickford to consult mid-shootout.

Suddenly, the country for whom shootouts were a national neurosis found itself winning them. On Saturday, the Three Lions scored with all five of their clinical penalty kicks, giving Southgate his third shootout victory in four attempts. The trouble is that the lone exception happened to be the final of Euro 2020 against Italy.

“We think we’ve got a good process," Southgate said. “We’ve been in four, we’ve won three. Of course, we got absolutely crucified for the one we lost. That’s always going to be the case."

Even with that defeat, Southgate’s record makes him undoubtedly England’s best-ever manager. He has won some 60% of his matches in charge and turned tournament semifinals into the bare minimum expectation. His tenure has made it easy to forget that until his first World Cup, in 2018, they had gone 12 years without winning a single knockout game.

In that period, there were tournaments the Three Lions missed altogether—and others they probably wished they had, such as Euro 2016, where they suffered the ignominious round-of-16 loss to Iceland. England, the inventor of the game, spent its time bumbling around major competitions, finding new ways to embarrass itself.

Under Southgate, England has now won eight knockout games at Euros and World Cups. In the previous 40 years, when tournaments admittedly had fewer teams, it had won just six.

“I took this job to try and improve English football," Southgate said. “I wanted us to regain credibility on the world stage."

On that front, Southgate has succeeded. England today is far from the laughingstock it was when it lost to Iceland. But rising expectations have ratcheted up the pressure on Southgate to the point where it seems unlikely he will remain in his post after the Euros, regardless of what happens now. That’s why Southgate made sure to savor the moment on Saturday and join his players in celebration on the pitch.

“Every now and then," he said, “you think, surely, there has to be some enjoyment in this job."

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