The two teams could not be separated over the course of 120 minutes, with Bukayo Saka on the scoresheet for England after Breel Embolo had sent Switzerland into the lead.
The contest went to penalties, with England proving to be perfect from the spot, scoring through Cole Palmer, Jude Bellingham, Saka, Ivan Toney and Trent Alexander-Arnold to advance to the semi-finals, as Three Lions goalkeeper Jordan Pickford had kept out an effort from Manuel Akanji.
England will now take on Netherlands in Wednesday night's semi-final for the right to play in next weekend's final against either France or Spain.
"I just thought the players were brilliant. It's the best we have played," Southgate told BBC Sport. "I thought we caused them a lot of problems. They are a really good side. They are hard to press, they are hard to defend against, their movement is good.
England will now meet Netherlands in the semi-finals
"To come from behind again and show the character and resilience we did, talking to the players about that, winning tournaments isn't just about playing well. It's not just about that. You've got to show other attributes to win and we showed them all tonight.
"We played well today and we had to be tactically spot on. I don't know what people think of us but we are in a third semi-final, so that says a lot about the group.
"We knew we needed to get pressure on their back three and that's not easy without changing the shape. I thought the players were disciplined with it.
"I thought Phil [Foden] was a real problem for them to pick up, Kobbie [Mainoo] and Jude were finding those gaps on the side of their block. I thought we had good control. For some of the balls into the box, we probably needed more men in the box, but l thought it was the best that we have played."
Southgate hails temperament of players
Southgate was also full of praise for the players that stepped up in the shootout, with Saka, who missed against Italy in the Euro 2020 final, among those to register.
"So brave. He is one of our best, so we were never in any question that he was going to take one. But we all know what he went through," he added.
"To deliver as he did.... but not just him, Trent and Ivan, to come on and take them as they did; Cole, he's like an old man in the way he is fearless. So a huge result for us, a huge performance, and we're still in it.
"We played well today and we had to be tactically spot on. I don't know what people think of us but we are in a third semi-final, so that says a lot about the group."
England have still only won one of their five games at this summer's European Championship in normal time, which proved to be a 1-0 success over Serbia in their competition opener.