After the 120 minutes finished level at 1-1, goalkeeper Jordan Pickford was the Three Lions hero in the shoot-out as he saved Switzerlands first spot-kick from Manuel Akanji.
Cole Palmer, Jude Bellingham, Bukayo Saka, Ivan Toney and Trent Alexander-Arnold converted their penalties to send England into a semi-final meeting with Turkey or the Netherlands in Dortmund on Wednesday.
After needing Bellinghams 95th-minute equaliser before going on to beat Slovakia in extra time in the last 16, England were again staring at defeat with 10 minutes to go.
A dreary encounter for the first 75 minutes came to life after Breel Embolo put Switzerland in front.
Sakas fine strike from outside the box levelled with Englands first shot on target five minutes later.
