South Africa came well and truly a cropper against Kane Williamson's cool and Colin de Grandhomme's muscle, New Zealand all but knocking their opponents out of a World Cup for a third time with a four-wicket win at Edgbaston.

Williamson and de Grandhomme shared in a 91-run stand that set up New Zealand's pursuit of South Africa's 241 for 6, Williamson pumping a last-over six off Andile Phehlukwayo that brought up a match-winning hundred before his umpteenth dab to third man brought up the win.

WATCH on Hotstar: Highlights of New Zealand's win over South Africa

South Africa spilled several chances in the field - and would have had Williamson caught behind in the 70s had they used their review - wilting under the pressure as New Zealand secured their fifth World Cup victory over them.

South Africa's scarcely believable lapse in gifting Williamson a life came in the midst of a haywire couple of overs when the match was in the balance, panic setting in and Faf du Plessis' men, unfortunately, were left gagging, coughing, and gasping for breath.

When Imran Tahir came on to bowl his final over, New Zealand were five down, needing a further 72 from 12 overs. Something, one felt, was about to give, and with his first ball Tahir very nearly had de Grandhomme caught by David Miller at full stretch close in on the leg side. Two balls later, another uppish whip flew hard to Miller's left, and he put in a dive that was no less hearty, but equally ineffective as he could only palm the tough chance away.

But worse was to come. Tahir's final delivery - a legbreak - gripped and spun past a Williamson, wrong-footed in a rare moment of uncertainty, flicking the finest of bottom edges on the way into Quinton de Kock's gloves. Tahir was convinced - as he always is - but de Kock didn't even appeal. South Africa had a review, and could have used it, but didn't.

The chances kept coming, but South Africa's luck had turned and with every missed opportunity, New Zealand pulled further ahead. They missed Williamson, again, when he had 77 and fended at a short one from Kagiso Rabada, hesitating before setting off on a run even as de Grandhomme sprinted across from the other end. Rabada swooped in his follow-through and shied with Miller in position for the run out, but Miller couldn't take it cleanly, disturbing the stumps empty-handed as the ball burst past him.

Two balls later, de Grandhomme edged Rabada through the vacant slip region to raise the fifty stand, leaving New Zealand needing 52 from 48.

While Williamson eventually sealed the deal, and bagged the Player of the Match award, de Grandhomme's knock was no less crucial in the final analysis. On a day when just one South African batsman was able to strike it at better than a run a ball, when everyone else struggled to come fully to terms with a two-paced track, de Grandhomme middled absolutely everything.

Born in Harare and playing for New Zealand at Edgbaston, his home away from home (he plays for Birmingham Bears in the Vitality T20), he swatted and clubbed his way to a 39-ball fifty that shattered South Africa's resolve.

When South Africa dropped short, he pulled powerfully. Full outside off, and he slashed. On his legs, he clubbed. It wasn't the most cultured knock - and he enjoyed a couple of lucky getaways along the way - but it was the perfect foil to Williamson's calm accumulation and helped to spark South Africa's anxiety. As the match slipped from their grasp, so did more chances - Williamson was dropped again (albeit off a high full-toss that was called a no-ball), and South Africa then contrived to turn an opportunity in the outfield off a de Grandhomme slog into four runs, Miller and Rabada both diving past the ball.

Williamson, meanwhile, was anything but anxious. New Zealand's captain is rarely better than when constructing a chase, and he passed 1000 ODI runs in England (at a monstrous average of 74.28 while systematically cobbling together his 19th fifty-plus total, and fifth hundred - in one-day chases.

There's no violence to Williamson's batting, he bats utterly without ego and is clearly mighty effective all the same. While he was pristine on the drive all afternoon, the shot that sticks in the memory is the dab down to third man. No less than 21 of his runs came in that region, with a total of 32 coming behind square on the off side. Fittingly, it was also the shot with which he sealed the result in the final over, one ball after he had brought the scores level by rocketing his first six - and first shot in anger - in an echo of Grant Elliott's dream-shattering knock during the last World Cup.

New Zealand's World Cup record against South Africa now stands at 6-2, while Williamson completed what was just the sixth century from a captain in a successful World Cup chase.

It could all have been very different, had South Africa been able to find inspiration in Chris Morris' breakthroughs earlier in the innings. Following on from the lucky dismissals of Colin Munro (caught and bowled off an inside edge on to his thigh) and Martin Guptill (hit wicket), Morris burgled a wicket in his fourth over when he strangled Ross Taylor down the leg side with a ball that could easily have been glanced for four.

There was nothing lucky about his next wicket, a brutal lifter finding Tom Latham's outside edge on the way through to de Kock, and at 80 for 4, it was anyone's game. Morris returned in the 33rd over with the match once again in the balance, snapping a 57-run stand when James Neesham steered one to slip. But then came Williamson and de Grandhomme's partnership, and whatever luck South Africa had enjoyed evaporated.

They might have been able to manufacture a little more of their own good fortune had they not been quite so tentative with the bat earlier in the day. Admittedly, there must have been some trepidation when they lost a rain-delayed toss and were put in this morning, and the early loss of de Kock to Trent Boult (again) won't have helped.

As a result, South Africa were hesitant up front, and equally diffident when the time came to swing the willow at the death. For much of their innings, it seemed like they had turned their back to the future and were stuck in 1990s mode: Hashim Amla ground out the third slowest fifty of his career, sharing in plodding fifty stands with du Plessis and Aiden Markram that essentially bought South Africa time without really taking the innings anywhere, and the top four's strike rates were all in the 60s.

South Africa finally found some momentum when Rassie van der Dussen was joined by Miller, the pair adding 72 for the fifth wicket at close to a run a ball. They took a while warming up the engine, and had hit just one boundary between the two of them by the time the fifty stand came up in the 43rd over, but van der Dussen gave the innings a little flourish at the finish with 15 runs off the final over - and he was the only batsman to finish with a strike rate above 100.

He ended with 67 not out, and South Africa reached 241 for 6, but it was not to be. Williamson's match-winning hundred capped New Zealand's day, and South Africa's World Cup dreams are all but extinguished.