Fazalhaq Farooqi's run out ended Afghanistan's chance to win Wednesday's game in regulation with a ball to spare, leaving both teams tied on 187.
Afghanistan posted 17 in the first super over and Farooqi was in position to win it again, restricting South Africa to 11 runs with one ball remaining until Tristan Stubbs plundered a six to level the scores again.
South Africa batted first in the second tiebreaker, posting 23 with David Miller and Stubbs combining for three sixes.
Keshav Maharaj gave South Africa control with a wicket and two dot balls to start the second super over, leaving Afghanistan needing 24 from four balls — or four sixes.
Enter Rahmanullah Gurbaz. With nothing to lose, the big-hitting opener - who earlier blazed 84 from 42 as Afghanistan chased South Africa's target of 6-187 - hit three consecutive sixes to reduce the requirement to six off one delivery.
A wide from Maharaj lowered it to five from one ball and raised the prospect of yet another tiebreaker.
But the South Africa spinner rebounded by having Gurbaz caught at backward point on the next delivery to end the game.
Lungi Ngidi, who took 3-26 during the match then bowled the first of the super overs, was voted player of the match.
"I've lost so much weight today. I've never been that stressed in my life in a cricket game," he said. "Being able to win two super overs with our hitters hitting like that … very happy."
South Africa is 2-0 after winning their opening game over Canada.
Afghanistan is 0-2 after an opening loss to New Zealand and unlikely to progress to the Super Eights from a group containing three highly-ranked teams.
Only the top two teams in each of the four groups will advance.
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