With three wins out of as many games, West Indies thus became the second team after New Zealand to enter the semi-finals of the marque event.
South Africa are virtually out of the tournament and their fate now hinges on the result of tomorrow's (March 26) Super 10 Group 1 match against England and Sri Lanka at New Delhi.
If England win tomorrow, Eoin Morgan's men will secure their place in the semi-finals as the second team from Group 1 and the match between South Africa and Sri Lanka will be a mere formality.
Electing to bowl, West Indies first restricted South Africa to 122 for eight and then chased down the target with two balls to spare with three wickets in hand.
Chasing 123 for win, West Indies were dealt a severe blow in the fifth ball of their innings when dangerous Chris Gayle was cleaned up by a late in-swigning delivery from Kagiso Rabada.
Johnson Charles (32) took the offensive to the opposition, hitting Chris Morris for two boundaries to pick up 11 runs of the fourth over.
Known for their fielding prowess, South Africa stayed on in the contest, courtesy a brilliant effort from Rilee Rossouw whose's direct throw on to the stumps caught Andre Fletcher (11) short of the crease.
Charles carried on his good work in the company of Marlon Samuels (44) before the former went for a huge swipe only to give a skier to South Africa captain Faf du Plessis at cover.
But Rabada's next over eased the pressure for West Indies as he went for three boundaries - one by Dwyane Bravo and two by Samuels.
Bravo, however, departed in the last delivery of the next over handing a simple catch to Wiese at backward point off Aaron Phangiso while going for a wild slash outside off-stump.
With West Indies needing a manageable 24 off 24 balls, it took a superb 17th over from leg-spinner Imran Tahir (2/13) to bring South Africa back into the match.
After dropping a difficult chance from Samuels off his own bowling, Tahir came back strongly and picked up two wickets in consecutive balls to the delight of the Proteas.
Tahir first dismissed Russell as the batsman went for a huge slog across the line only to find Miller at deep midwicket and then castled West Indies skipper Daren Sammy with a first ball googly.