Alexander-Arnold, dropped from the starting team after two disappointing performances in the group stage, came on as a late extra-time substitute to take the glory after Jordan Pickford had saved from Manuel Akanji.
It was the third successive Euro 2024 quarter-final to go to extra time, two of them to penalties.
Breel Embolo had put Switzerland ahead after 75 minutes, with Bukayo Saka levelling five minutes later in a game of few chances.
Saka, who missed in England's Euro 2020 final shootout defeat by Italy, stood up again on Saturday and converted confidently in front of a wall of England fans.
"He is so brave, he's one of our best so we were never in any question that he was going to take one," coach Gareth Southgate said of the midfielder.
Saka said: "To come back from something like that was really difficult. I used that to make me stronger. Today I took the chance and I'm happy.
"Hopefully next game we can win in the 90 minutes but if this is what it takes, we'll do anything."
That next game is Wednesday's semi-final in Dortmund against either Netherlands or Turkey, and though England will almost certainly have to find more intensity and accuracy in attack than they showed on Saturday, the confidence gained from another shootout win after decades of pain will be priceless.
"Incredible," said Alexander-Arnold. "These are the goals and aims that we set for ourselves. The team showed a lot of character, belief, a lot of heart and spirit out there. We dug deep when it mattered most.
"A lot of practice goes into that moment," he added of the penalty he smashed into the top corner. "I practice. I knew what spot. I knew I just needed to execute it and I went out and did that."
England came into the game on the back of two dire performances and, though they showed more invention and movement there was again little end product, and with the Swiss equally cautious, there was barely a meaningful effort on goal for an hour.
Switzerland, impressive victors over Italy last week and seeking their first major semi-final, took the lead after 75 minutes when Dan Ndoye whipped in a deflected low cross from the right, with Breel Embolo stretching in front of Kyle Walker to poke it home.
Southgate responded by immediately throwing on three substitutes – Cole Palmer, Luke Shaw and Eberechi Eze - and five minutes later England were level as Saka cut in from the right wing and curled a brilliant 20-metre left-footed shot in off the far post.
In extra time England's Declan Rice had a fizzing shot from the edge of the box brilliantly saved by a diving Yann Sommer, while Xherdan Shaqiri clipped a post direct from a corner, before the penalties that had felt inevitable almost from the start duly arrived.
Switzerland had lost three of their four major tournament shootouts, while England’s famously uncomfortable record was seven defeats and two wins.
That reads 7-3 now after Cole Palmer, Jude Bellingham, Saka, and Ivan Toney all scored confidently ahead of Alexander-Arnold's finale.
"Huge performance, huge result for us," Southgate said.
"We had to be tactically spot on. To come from behind again and show the character and resilience that we did... winning tournaments isn't just about playing well. I thought we did play well today but you've also got to show all those other attributes to win and we showed them all tonight."
The Swiss also lost in the quarter-finals of the last Euros on a shootout, and Shaqiri said: "There is nothing more brutal than to go home after penalties.
"We fought and gave it all we had to try and qualify so that the fairytale could continue.
