Murat Yakin's men were on course to reach the last four thanks to a Breel Embolo tap-in deep into the second half, but Bukayo Saka responded with a stunning effort just five minutes later to force an additional 30 minutes.
Neither side could find an extra-time winner - despite a magnificent Xherdan Shaqiri corner which hit the upright - but while Manuel Akanji missed his spot kick for Switzerland, England dispatched all five of their efforts from 12 yards to dramatically reach the last four.
Here, Sports Mole provides player ratings from England's unmissable triumph in Gareth Southgate's 100th game in charge.
England player ratings vs. Switzerland: Saka takes Man of the Match prize
GOALKEEPER
Jordan Pickford - 8/10
Not a lot he could have done with Embolo's goal and had little else to do in the 90. Nearly caught out by Shaqiri's wonder corner but came up with the critical save in the shootout, as he so often does for his country.
DEFENCE
Kyle Walker - 6/10
Allowed Embolo to get in front of him too easy for the Swiss attacker's goal and was required to defend for most of the game, inhibiting his ability to get forward.
Ezri Konsa - 8/10
Selected to deputise for the suspended Marc Guehi and made a number of crucial blocks and clearances in the first half. Largely won his battle with Embolo as well. Impressive stuff from the Aston Villa man.
John Stones - 6/10
Should have done more to keep out Dan Ndoye's cross for Embolo's strike, which took a nick off of his knee en route to the Monaco man. But was strong in the air throughout.
MIDFIELD
Kieran Trippier - 5/10
Struggled to cope with the pace and directness of Ndoye for the majority of his time on the pitch, despite getting stuck in quite a bit. Minimal attacking threat and was replaced just after the opener.
Declan Rice - 8/10
Another controlling midfield performance from the Arsenal maestro, who misplaced just one pass throughout the entire contest and registered the assist for Saka's goal. Not afraid to stick his foot in either, as is typical with the omnipresent £105m man.
Kobbie Mainoo - 7/10
Another highly industrious display in the first half - both offensively and defensively - and was only denied an opener by a brilliant Granit Xhaka block. Faded slightly in the second before being taken off.
Bukayo Saka - 9/10
Star. Boy. Will be living in Michel Aebischer's nightmares for weeks to come, spared England's blushes with a wonderful equaliser and could not have been braver to step up to the penalty spot in the shootout. A deserved man of the match.
ATTACK
Phil Foden - 6/10
Much, much more comfortable in an inverted position and saw plenty of the ball in the final third, but was still overshadowed by the likes of Saka and Mainoo. Just one shot and one chance created.
Jude Bellingham - 7/10
Grew in influence at the first half went on and went on some mazy runs. Another confident penalty in the shootout and one of his better overall displays.
Harry Kane - 4/10
Anonymous for the most part in the first half - should have been in the box to receive an early Saka cutback - and booked in the second for a somewhat innocuous elbow on Manuel Akanji. This England system has just not been working for him this tournament. Taken off in extra time after partially clattering into Southgate.
SUBSTITUTES
Eberechi Eze - 6/10
Typically energetic after being brought on but did not have the biggest impact in the final third.
Cole Palmer - 7/10
Completely unfazed by stepping up first in the shootout and did what he does best from the chalk. Not half bad on the left in extra time either.
Luke Shaw - 7/10
He lives! A first England appearance in over 12 months for the left-back, who provided the overlapping runs that England have so lacked. Surely in contention to start the semi-final?
Ivan Toney - 7/10
Toney takes penalty, Toney scores penalty. Little else needs to be said.
Trent Alexander-Arnold - 7/10
Played just five minutes in extra time but had ice running through his veins as he struck the spot kick that sent England through.
Switzerland player ratings vs. England: Akanji crushed after impressive overall display
GOALKEEPER
Yann Sommer - 6/10
Could have at least tried to dive for Saka's beautiful finish but may have been inconsequential either way. Made a top stop from Rice in extra time but was powerless against the perfect England penalties.
DEFENCE
Fabian Schar - 7/10
Booked in the first half for a foul on Bellingham, but could count himself unfortunate in all honesty. Played a huge role in the goal with a slick pass into Ndoye and scored from the spot, but maybe lucky not to be sent off.
Manuel Akanji - 6/10
The penalty miss was the most sour of sour notes for the Manchester City man, who otherwise did his bit to help keep Switzerland's defensive structure in the 120 minutes and won four of his five duels. None of that matters now, though.
Ricardo Rodriguez - 7/10
Made a terrific early block to deny Mainoo's long-ranger and was potent in both thirds of the field for Switzerland. Hard to believe he is still only 31.
MIDFIELD
Remo Freuler - 6/10
Somehow avoided giving away a free kick for a blatant handball in the first half, but played a nice ball through to set up Breel Embolo's early second-half chance.
Granit Xhaka - 7/10
Kept Switzerland ticking over early and denied Mainoo just before the break with an amazing sliding tackle. His influence did not wane in the second 45 either.
Michel Aebischer - 4/10
Absolutely no match for Saka and was dribbled past five times in the first half. Was accurate with his passing at least, but he will want to forget this one as soon as possible.
Fabian Rieder - 5/10
Held his place in the XI despite Silvan Widmer's return from suspension but was quiet. One shocking delivery from a corner shortly before he was taken off.
ATTACK
Dan Ndoye - 8/10
Had the beating of Kieran Trippier all day long and bagged the assist for Embolo's goal, even if it took a slight deflection on its way through. Can see why Manchester United reportedly want him.
Ruben Vargas - 5/10
Could not replicate his wonder goal against Italy and only had one or two lively moments in attack. Taken off just after the hour.
Breel Embolo - 7/10
Had the game's first shot on target after 51 minutes, but it was weak. Largely lost his battle with Konsa but easily outmuscled Walker to prod home. He always finds a way in major tournaments.
SUBSTITUTES
Silvan Widmer - 6/10
Back from suspension and was a solid force on the wing after coming on, but nothing spectacular.
Steven Zuber - 6/10
Was heavily involved in most of Switzerland's late attacks but did not have a clinical edge.
Denis Zakaria - 6/10
Always nice to see him on the pitch but could not get involved in his late cameo.
Xherdan Shaqiri - 6/10
Wasted a Switzerland counter with an awful attack before almost catching Pickford out with an unbelievable corner, which nearly crept in at the near post. Scored in the shootout.
Zeki Amdouni - 7/10
Forced Pickford into a good save in his two-minute cameo and scored from the spot to keep Switzerland in the quarter-final before Alexander-Arnold did the rest.
Vincent Sierro - N/A
Came on in the 118th minute and touched the ball three times.