Breel Embolo had put Switzerland in front with 15 minutes of normal time remaining, leaving the Three Lions in huge danger of crashing out, but Bukayo Saka stepped up almost immediately to fire home a brilliant equaliser.
It remained level after 120 minutes, the quarter-final settled from the spot.
Jordan Pickford saved Manuel Akanji’s penalty, Switzerland’s opener, and England were perfect from there, culminating in Alexander-Arnold, a late substitute in extra time, dispatching his penalty in emphatic fashion.
It was a dramatic end to an edgy encounter and England's first shoot-out win since beating the Swiss in the Nations League third-placed play-off five years ago in Portugal.
Pickford had a decisive say then, just as he did on Saturday night as Gareth Southgate's 100th match ended with a victory that sets up a Dortmund clash against Turkey or the Netherlands on Wednesday.
Switzerland stuck with the same side that knocked out holders Italy as Southgate made one enforced change and a big switch in system.
Ezri Konsa replaced suspended centre-back Marc Guehi and formed part of a back three, with eyebrows raised at the decision to deploy Saka as right wing-back rather than on the opposite side.
Southgate's side made an encouraging start with crisper passing and better pressing than previous matches as Declan Rice and Mainoo saw shots blocked.
Phil Foden roamed more effectively and Jude Bellingham - cleared to play after making a gesture after scoring against Slovakia - probed, with Saka giving Michel Aebischer a torrid time. His lively dart into the box just before the break led to Mainoo being denied by a great Granit Xhaka tackle.
The half ended without stoppage time or a shot on target from either side but within six minutes of the restart Embolo got a shot on Jordan Pickford's goal.
The Swiss striker saw a headed effort blocked by Konsa as Murat Yakin's improved side threatened with increased regularity as they attacked their bouncing sea of red.
England fans turned up the volume in response, only to be silenced in the 75th minute. A smart Swiss passing move saw Newcastle's Fabian Schar slip in Dan Ndoye to send a low cross that John Stones could only help onto the far post, where Embolo held off Kyle Walker to score.
The England boss immediately brought on Cole Palmer, Eberechi Eze and Luke Shaw - making his first appearance since mid-February - after a goal rocked that England but did not shake their spirit.
Just five minutes after falling behind, Saka confidently cut in from the right and arrowed a low left-footed shot from the edge of the box that beat Yann Sommer, coming off the inside of the far post, and sparked celebrations across the land.
The feeling was as much of relief as joy but there was still an anxious moment as a superb Schar cross was taken off the head of Ndoye by Embolo with an open goal gaping.
Rice went agonisingly close to putting England ahead five minutes into the first period of extra time, hitting a brilliant 25-yarder that Sommer tipped wide.
The Swiss goalkeeper plucked a Bellingham curler out of the air and Southgate replaced skipper Kane with Ivan Toney shortly after he was bundled into the dugout.
England were being forced back and Xherdan Shaqiri sent an audacious whipped direct effort from a corner onto the crossbar in the 117th minute. Pickford stopped a Zeki Amdouni shot and a stretching Silvan Widmer turned over as England lived dangerously.
Toney was bundled over in the box but there was no penalty, just a shoot-out.
Spot-kick king Cole Palmer beat Sommer to set the tone before Pickford stopped Manchester City defender Akanji low to his left. Every other penalty taker scored, most notably Saka, and substitute Alexander-Arnold's effort sent the hordes in white wild.
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