Kane Williamson gets second century as New Zealand build lead of 528
The hosts will resume on 179-4 on day four after deciding not to enforce the follow-on on the inexperienced Proteas despite holding a first-innings lead of 349 runs when they bundled the tourists out before tea.
Williamson's 109 from 132 deliveries on another sun-kissed day at Mount Maunganui made him only the fifth New Zealander to score a hundred in each innings of a test.
The former Black Caps captain came to the crease after Dane Paterson had removed opener Tom Latham lbw for three in the fifth over of New Zealand's innings.
Scoring at a pace that appeared beyond the other batsmen, Williamson brought up his 34th test half century off 75 balls and his 31st century with a single through the covers off the 125th delivery he faced.
As in the first innings, he had a life courtesy of Edward Moore halfway through his innings when the South African dropped a sitter off Paterson at backward square leg.
Moore made amends in the next over with a brilliant diving catch to remove Devon Conway for 29 off the spin of South Africa's test debutant captain Neil Brand.
A Brand catch off a Ruan de Swardt delivery ensured there would be no second-innings heroics for Rachin Ravindra, who scored 240 in the first but departed for 12.
Williamson pummelled the first six of his innings after reaching the century mark but was stumped after advancing down the track in search of another five balls later to give Brand his eighth wicket of the match.
Daryl Mitchell will resume on day four unbeaten on 11 with Tom Blundell five not out alongside him as New Zealand look to make an all but unassailable position even stronger.
"It's been a tough couple of days, I think we've seen enough of Kane now," said South Africa's Keegan Petersen.
"We've just still got to believe that we can get something out of this game."
The tourists, boasting only 51 caps between them as Cricket South Africa prioritises their Twenty20 competition (SA20), resumed on 80-4 and added only three runs to their tally when seamer Matt Henry struck with two wickets in three balls.
Petersen's knock of 45 extended his team's innings beyond the lunch break but all-rounder Ravindra had him caught off a rash shot and it was then just a case of mopping up for New Zealand.
"I think (we wanted to) give the big boys a bit of a rest and see if this wicket slows down or cracks up a little bit more," said Mitch Santner, explaining why the follow-on had not been imposed.
"Hopefully it starts to play some tricks," added the spinner, who took 3-34 in South Africa's first innings.
New Zealand have never beaten South Africa in a test series in nearly a century of trying. The second and final test in the series takes place in Hamilton from February 13.