The result leaves India at the top of the group with seven points, one ahead of West Indies and South Africa, with England on five. Co-hosts Bangladesh are on four points with a game in hand on the English.
Four teams from the group will go through to the quarter-finals but no team has yet booked its place in the last eight.
Needing 13 to win in the last over off seamer Ashish Nehra, South Africa found an unlikely star in Robin Peterson who plundered 16 runs off the first four balls to help his side achieve a 297-run target and a three-wicket win.
India earlier looked set to post a huge total when they reached 267-1 following Sachin Tendulkar's brilliant 111 for his 48th one-day hundred, but lost their last nine wickets for just 29 runs to be bowled out for 296.
Man of the match Dale Steyn (5-50) played a key role in his team's fightback to bag his maiden five-wicket haul in one-day internationals.
