Kagiso Rabada took 6-32 in the second innings for 13 wickets in the match for the 20-year-old South African fast bowler, only the third South African to take 13 in a Test.
Continuing on the final day on 52-3, there was little resistance from England at SuperSport Park.
James Taylor and Joe Root fell in the first four overs of play, and England lost seven wickets in 13 overs for 49 runs on the day.
For South Africa, it was a morale-boosting win for new captain AB de Villiers and his team, which was knocked off the top of the Test rankings with the series loss.
It was the Proteas' first Test win in its last 10 Tests.
Rabada, in just his sixth Test, was the biggest plus for South Africa in Centurion, bowling with pace and getting seam movement and swing to trouble England in both innings.
Starting the final day, England faced a victory target of 382, way more than the current record at SuperSport Park, or more realistically three sessions to survive for a draw.
England's two overnight batsmen fell in the space of four balls and after adding just six runs to the total.
Rabada's first 10-wicket haul in Tests came in strange fashion: Jonny Bairstow was caught behind prompting a big celebration, only for a TV review to reveal Rabada had overstepped for a no ball. Next delivery, Rabada sent in a ball that shaped away to take an edge and remove Bairstow straight after his reprieve.
Ben Stokes, England's star of the series, could do little, hitting two quick fours for his 10 before being caught in the deep.
England had lasted barely an hour of the fifth day.
Rabada became just the third South African after Hugh Tayfield and Makhaya Ntini to take 13 wickets in a match. His final figures of 13-144 were the second best for South Africa behind Ntini's 13-132 against West Indies in 2005.
As Rabada took wickets on the final morning, Ntini, now a commentator in South Africa, shouted "Come on Rabada, get 13!" from the stands.
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