Faf du Plessis (51 off 90 balls) and David Miller (45 off 35 balls) saw the visitors home with an unbeaten 62-run fifth-wicket partnership.
New Zealand's chances blossomed when Jimmy Neesham delivered one of the few balls, possibly the only ball this series, which de Villiers took his eyes off. Wicketkeeper Luke Ronchi kept it in his sight. De Villiers exited for 23 from 21 balls, his lowest score of the series.
The wickets fell steadily - Quinton de Kock in the third over, Hashim Amla in the 12th and Jean-Paul Duminy in the 16th - but the sight of de Villiers loosening the velcro on his gloves gave locals hope. It wasn't enough.
Earlier, New Zealand's top order batting got as scorched as the straight six from Neesham which flew through the flame throwers in the 21st over.
The leather withstood the heat and came back unscathed; the hosts could not claim the same invincibility against the constant needling outside off stump.
South Africa's bowling pyrotechnics were impossible to keep at bay. A gauge of their effort came with the crowd's ironic applause for singles towards the end of the 41.1 over ordeal. The key period was the collapse from 41 for one at the start of the 11th over, to 72 for five by the end of the 20th.
The hosts faced the indignity of a delayed dinner break to meet allotted broadcasting times.
Kagiso Rabada (three for 25 from 7.1 overs), Andile Phehlukwayo (two for 35 from seven overs), Chris Morris (one for 34 from nine overs) and Imran Tahir (two for 14 from his allotment) worked in partnerships to ensure no miraculous Black Caps escape.
Tahir's figures were second-equal best for economy rate from 10 overs in Eden Park ODIs. They were level with the match's television umpire Paul Reiffel in 1995 for Australia.
Continuing the serendipity, Indian match referee Javagal Srinath is the king of parsimony from a full quota of overs at the ground, taking three wickets for 13 in 2003.
"It was far from easy, but that's when fight needs to be shown to get a competitive score," captain Kane Williamson said.
"It was tough to rotate the strike out there under pressure. When you do lose wickets that is something you look to do to bring momentum back, and that wasn't happening.
"We needed two guys to stick there for a 'death phase' so we could go harder with wickets in hand."
Williamson was run out for nine at the end of Tahir's first over as the dot balls mounted.
Dean Brownlie (24 off 37 balls) and Ross Taylor (eight from 12) fell in identical lbw fashion to Phehlukwayo; hit on the back leg playing across the line. Ronchi had a chance to redeem his recent poor run of ODI form, but gloved a Morris delivery to de Kock for eight off 20 balls.
Neesham, with 24 from as many deliveries, provided glimpses of clean hitting while Colin de Grandhomme (32 from 48 balls) and Mitchell Santner (24 from 50 balls) contributed the best partnership of the innings with 45 for the seventh wicket. Santner was casually run out by Duminy as he eased a ball behind point.
The rout was complete when de Grandhomme edged Rabada behind while attempting to swing him through the legside.
