Handre Pollard's penalties kept South Africa in touch — the sides turned around with the score 17-12 at halftime — and the Boks were dominant after the break, with a change of their entire front row in the 45th helping to turn the tide.
The English finished the game having conceded 18 penalties, with one of them allowing replacement Elton Jantjies to boot a penalty to put South Africa ahead for the first time in the 64th.
Within a minute, England was back in front and it was another try that owed to the team's creativity in the backs as Henry Slade sent Joe Marchant through a gap and he passed inside for replacement scrumhalf Raffi Quirke to sprint over for his first international try.
It was soon advantage South Africa when replacement prop Will Stuart was yellow-carded for an accumulation of team offenses and the visitors capitalized, with Makazole Mapimpi romping over in the left corner for their only try and Steyn then kicking a penalty for 26-24.
There was still time, though, for another big turning point when Kolisi attempted to challenge for a high ball with Marchant. He was beaten to it and ended up pulling down England's winger while in the air.
Kolisi protested but he had to go to the sin-bin. England launched an attack, got a penalty advantage, but still continued going forward and earned a penalty in a much better position after Steyn slid, knee-first, into the back of an England player on the ground.
Smith made no mistake again and England managed to claim the restart to see out a brilliant test, completing a sweep of autumn victories after previous wins over Tonga and Australia.
"We dug in, we fought," said Steward, who had to field a stream of high kicks at fullback. "South Africa are so physical and you can't shy away from that.
"It was a whole squad effort and it was unbelievable."
- AP