The Springboks led 13-12 at half-time only to be dominated after the break and finished with 13 players after Marco van Staden and Cobus Reinach were yellow-carded.
Will Jordan scored two tries and fellow wing Josh Moorby, lock Fabian Holland and flanker Luke Jacobsen one each for the All Blacks.
Fly-half Ruben Love kicked four conversions.
The lone South Africa try came from substitute back Andre Esterhuizen.
Wing Cheslin Kolbe kicked the conversion and three penalties.
Second-ranked New Zealand gave the Rugby World Cup holders a lesson in taking chances with South African making numerous entries into their rivals’ 22 without any reward.
South Africa led after five minutes when Kolbe — preferred to fly-half Sacha Feinberg-Mngomezulu as the goal kicker — landed a close-range penalty.
New Zealand then edged in front four minutes later as Jordan, making his first tour appearance after recovering from injury, eluded Grant Williams to score in the corner.
Springbok wing Kurt-Lee Arendse was forced to retire after failing a head injury assessment and his replacement, Esterhuizen, put the home team back in front on 14 minutes.
The hybrid player — he can operate in the backline and as a loose forward — shrugged off a tackle from Moorby to score and Kolbe converted for a 10-5 advantage.
But once again the All Blacks hit back with an offload from flanker Tupou Vaa’i sending Moorby clear to score and Love converted to put the tourists two points ahead.
Kolbe slotted his second penalty of the half soon after and South Africa nudged into a 13-12 lead that they retained until half-time.
New Zealand prop Ethan de Groot and lock Josh Lord had to leave the field just before the break after they collided and sustained head injuries.
Kolbe maintained his perfect goal-kicking record from another penalty seven minutes into the second half, but the match then took a dramatic turn.
After surviving falling further behind with desperate defending, New Zealand forwards Fabian Holland and Luke Jacobson scored tries.
Love converted both for a 26-16 lead.
Holland blocked a Williams clearance to score, then Jacobson barged over after a five-metre scrum.
As if suddenly trailing by 10 points was not sufficiently worrying, the Springboks lost Kolbe, who limped off to be replaced by Reinach.
Then came the yellow cards before Jordan scored a soft try in the closing stages after snatching a long pass from captain Ardie Savea. Love converted.
The second Test is in Cape Town on August 29. (AFP)
