The first test was a damp draw, but there was no chance of rain in the second to save the Black Caps.
The South Africans declared for a second time at 132-7 in the morning, setting New Zealand an unlikely 400 to win its first test against the hosts in 12 years, and first series against them ever.
But fast bowler Dale Steyn got rid of both opening batsmen in the first over, New Zealand was reduced to 7-4 after 19 balls, and was eventually all out for 195.
That gave South Africa the win for an 11th time in 14 series between them since 1932.
In taking his 26th five-for in tests, Steyn started and finished New Zealand's demise.
He removed openers Tom Latham and Martin Guptill for ducks, before Ross Taylor, who ran himself out for 1 in the first innings, was unlucky to fall for a duck to a Steyn ball that got him lbw on his ankle.
Fellow quick Vernon Philander took the prize wicket of New Zealand captain Kane Williamson (5) and, at 7-4, victory was beyond the visitors.
The inevitable was stalled for four-plus hours by Henry Nicholls, playing in his sixth test. He hit a career-best 76, and received able backup from BJ Watling (32) and Mitchell Santner (16). Doug Bracewell pitched in with a quick 30 but Nicholls was the last man out, caught on the long leg boundary off Steyn. His 140-ball knock, his second half-century, included a dozen boundaries.
Steyn, with eight wickets in the match, passed Wasim Akram to sit 11th on the list of all-time test wicket-takers. With 416 wickets, he was five behind South Africa record-holder Shaun Pollock.
Temba Bavuma made 40 not out as South Africa, resuming from 105-6 overnight, declared its second innings on 132-7, on a tough pitch to bat on. South Africa set up victory with 481 in its first innings.
