The South Africans resumed on 289-7 on Day 3 and were bowled out for 326 after an hour’s play.
England didn’t even make it to tea.
The second test starts Thursday in Manchester, where the Proteas will look to wrap up the series.
South Africa captain Dean Elgar said he couldn’t believe how quickly the match had finished, especially since around half of Day 1 was lost to rain.
“But I think I will when I wake up tomorrow with a hangover,” Elgar said.
“It was a team effort, there were a few standout performances but everyone had played their part.”
England had given itself an outside chance of victory by wrapping up South Africa’s tail quickly, with Stuart Broad taking a wonder-catch to remove Kagiso Rabada for 3 off the third ball of the day before taking the final two wickets.
Anrich Nortje was left stranded on 28 not out and then helped to run through the English batting order, claiming three wickets in just 10 deliveries in the middle of the innings.
“It was a hostile spell of bowling,” Elgar said. “He is an angry man and you don’t want to poke the bear.”