Chasing a stiff target of 298 runs in 43 overs under the Duckworth/Lewis rule in the rain-curtailed game, New Zealand crossed the line when Grant Elliot (84) smashed Dale Steyn over long-on for a six with just one ball to spare in a thrilling showdown.
Needing 12 to win off six balls, Vettori found the fence to bring the equation down to five from two balls, before Elliot hit the winning runs as the Eden Park broke into wild celebrations.
The result meant the perennial choker's tag would continue to haunt South Africa and they would once again look for soul searching.
Since the inaugural edition in 1975, New Zealand have reached the semifinals of the showpiece event six times, but failed to clear the penultimate hurdle on all the occasions.
However, they looked determined to script history today and despite being set a daunting target, the Kiwis never gave up and were spurred on by a vociferous home crowd.
Earlier, David Miller had blazed his way to a 18-ball 49 as the Proteas set a difficult target. Opting to bat, South Africa notched up 281 for five, but the target was revised under the Duckworth/Lewis method after seven overs each were deducted per side following a two-hour rain interruption.
Skipper AB de Villiers made an unbeaten 65 off 45 balls, while Faf du Plessis top-scored with a 107-ball 82, but it was Miller's innings that gave Proteas the much-needed impetuous towards the end.
In reply, the New Zealanders were off to a flying start with skipper Brendon McCullum going hammer and tongs at the South African pacers, especially Steyn.
Such was his onslaught against Steyn that the South African pace spearhead was clobbered for 25 runs as New Zealand raced to 71 for no loss in five overs.
McCullum's two sixes and three fours against Steyn was followed by a maiden over by Imran Tahir, and it worked in the Proteas' favour as the pressure created by the leg-spinner saw McCullum throwing away his wicket. His manic fifty contained eight boundaries and four sixes. The belligerent 26-ball 59 set the tone for what turned out to be a historic chase.
Elliott and Corey Anderson (58) played smartly, mixing caution with aggression, and it was their 103-run partnership for the fifth wicket that kept the Black Caps in the game after McCullum's dismissal.
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