New Zealand had taken a 10-7 lead into the break on the back of a frantic first half that saw tries from debutant Leroy Carter and Cheslin Kolbe, as well as a Damian McKenzie penalty.
However, Rassie Erasmus' half-time team talk did the trick, as his world champion charges came out firing in the second period and added tries through Kolbe's second, and further scores from Damian Willemse, Kwagga Smith, RG Snyman and Andre Esterhuizen.
The defeat breaks the record the Springboks held from their 35-7 victory over the All Blacks in Twickenham in the lead-up to the 2023 Rugby World Cup.
It also means South Africa avoided suffering back-to-back defeats for the first time since 2022, and are now second in the Rugby Championship standings led by Australia, who narrowly lost to Argentina earlier in the day.
There are now just two points separating all four teams in the Rugby Championship after each recorded two wins and two losses in their first four outings. South Africa sit second on points difference over the All Blacks, who are one spot ahead of bottom side Argentina.
"I give credit to my boys, we didn't give up. And to the coach as well," said Springboks captain Siya Kolisi.
"We played fearlessly, we didn't want to hold back. We knew what we needed to do and we believed in ourselves, that was the most important thing out there.
"We've got the right to fight for the Rugby Championship. It's open now and that's what matters to us."
New Zealand captain Scott Barrett said: "That's a tough one to swallow, The Springboks certainly showed up and took a real improvement from last week and we just didn't adjust. Well done to Siya and his team.
"There's a lot of areas where we can get better. So we'll make an adjustment heading forward."
All Blacks edge frantic first half
The Springboks were on the attack early in the game in Wellington, turning down points in favour of kicking to the corner within the first two minutes of a rollercoaster first half that did not provide the points it promised. Early inroads were not forthcoming as first Malcolm Marx missed his jumper at the ensuing lineout from the penalty, and shortly thereafter Snyman's short-range try was ruled out for a knock-on in the build-up from scrum-half Cobus Reinach.
In the end it was the All Blacks who were on the scoreboard first, with Carter marking his Test debut by getting on the end of a well-worked move down the left that exposed the Springboks' rush defence.
That momentum was short-lived, however, as Kolbe made good on that rush defence by intercepting a short pass on the same edge to race over untouched at the very same ground where he got his first Test try in the exact same manner in 2018.
That locked the scores up, but when South Africa failed to roll away at a ruck, Damian McKenzie stepped up to calm things down with a penalty that gave the hosts a 10-7 lead at the break.
Springboks flex their muscles
South Africa were a completely different beast from the restart, and went at it straight from the outset. From penalty advantage at a scrum, Reinach darted ahead before popping to Siya Kolisi, who raced up the field. The skipper was eventually brought down, but the ball was then worked down the right for Kolbe to step inside the cover defence and give the visitors the lead for the first time in the game.
Manie Libbok, who kicked superbly all night, converted Kolbe's effort and added a penalty to make it 17-10 to the Boks, who appeared settled by their early second-half success.
The South African forwards then began to squeeze their counterparts, earning a string of scrum penalties along the way to a stolen lineout on the opposition 5m line, from which recycled ball saw Willemse take a loose pass from Libbok to power his way over for a score that put the visitors 14 points clear.
The All Blacks were still in it at that point, but then Ethan Hooker countered brilliantly from distance, carving his way through the defence to get the Boks deep into enemy territory.
From there Libbok's cross-field kick found Pieter-Steph du Toit, who popped to the supporting Smith for a try that just about put the game to bed. Snyman was not to be denied, however, and cantered through from deep to spark the kind of celebrations that only come when you know a victory over the All Blacks is assured, before Esterhuizen put the finishing touches on the victory by muscling over the line after replacement Grant Williams had danced his way down the left.
South Africa are now in the pound seats to defend their Rugby Championship title, as they face bottom side Argentina in the last two rounds of the tournament in Durban and London, while New Zealand and Australia go head-to-head in a Bledisloe Cup double header in Auckland and Perth.