England then faced a penalty shootout without their captain and de facto penalty taker Harry Kane, who had been replaced by Ivan Toney in the second half of extra time.
There was no need for England fans to panic though, as Gareth Southgate’s side scored all five of their penalties.
After Cole Palmer stepped up first and scored for England, goalkeeper Jordan Pickford helped the team to an early advantage in the shootout as he saved Manuel Akanji’s strike.
England’s Jude Bellingham, Bukayo Saka and Ivan Toney then followed suit with three more goals, leaving Trent Alexander-Arnold to coolly put away the fifth.
The triumphant display came three years after England were denied the Euro 2020 title when they lost on penalties to Italy.
The Prince of Wales was among fans who cheered England on at the Dusseldorf Arena and could be seen out of his seat and punching the air after their 80th minute equaliser.
England will play their semi-final match on Wednesday and face the Netherlands, who beat Turkey 2-1 later on Saturday evening.
The match had seen Switzerland go 1-0 up in the 75th minute after a goal from Breel Embolo – but England levelled the score just five minutes later with a goal from Saka.
For the second match in a row, England then faced extra time.
Declan Rice came close to giving England the lead five minutes in to the extra half an hour of play, but his long-range effort was tipped around the post by Switzerland’s keeper, Yann Sommer.
England’s quarter-final fightback comes after they were almost eliminated from the tournament last Sunday, after falling 1-0 down to Slovakia – until Bellingham scored a stunning equaliser with seconds to spare.
Kane then scored the winner just minutes into the first half of extra-time, and Southgate’s side held on despite some tense moments.
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Gareth Southgate’s post-match reaction
Southgate hailed his side for showing “incredible composure” during the penalty shootout and added: “I’m so, so proud of them.”
The manager also heaped praise on goal and penalty scorer Saka – who was one of the players to miss a penalty at the Euro 2020 final.
“He kept delivering,” he said. “And we showed the patience to keep probing and trying to tire the opposition.
“As well as all of that, you have to show other qualities to win in tournaments and these players are showing unbelievable qualities of character and resilience. It’s not just about playing well.”
