All Blacks back-rower Ardie Savea won a decisive turnover in the dying minutes of his 100th Test to halt a late attack and ensure New Zealand extended their unbeaten streak in Auckland to 51 matches.
Slippery conditions meant the contest between the world's two top-ranked sides never shone as a spectacle, but there was no shortage of physicality.
"It was a gutsy performance. I'm just proud of the boys' efforts, we knew the Springboks were going to come here and give it to us," said Savea.
The reigning two-time world champion Springboks were desperate to end New Zealand's remarkable record at Eden Park, where the hosts haven't lost since 1994.
However, the visitors paid a high price for a slew of handling errors in a match they dominated in terms of territory and possession, scoring two tries to the home side's three.
South Africa captain Jesse Kriel said: "We obviously weren't up to scratch tonight, way too many handling errors and things that just weren't up to standard for us.
"But full credit to New Zealand, I thought they were really good and was a special way to celebrate an awesome player for them in Ardie."
New Zealand halted a four-match run of losses to South Africa as well as retaining their number one world ranking.
They also stayed top of the Rugby Championship standings after three rounds, ahead of Australia by a bonus point.
'Stand up'
The All Blacks opened the scoring in spectacular fashion in the second minute when Emoni Narawa fielded a Beauden Barrett cross-kick and evaded two defenders to score in the corner.
The game ended shortly afterwards for recalled wing Narawa, who limped off with a leg injury.
New Zealand doubled their lead to 14-0 when fullback Will Jordan scored from a sharp lineout move, running off an inside pass from back-rower Wallace Sititi.
Springboks fly-half Handre Pollard landed their only points of the half with a penalty goal to be 14-3 down at the interval.
Replacement Damian McKenzie extended New Zealand's lead by three points but the home team's defence finally cracked in the 62nd minute when South Africa's dominant scrum won a tighthead and hooker Malcolm Marx powered over.
Springboks forward Kwagga Smith was shown a yellow card for a deliberate penalty and New Zealand capitalised on their one-man advantage, sending replacement back Quinn Tupaea across.
The visitors kept the contest alive with six minutes remaining when replacement scrum-half Cobus Reinach scored.
Savea's ruck turnover deep on defence elicited a roar from the crowd of 48,000 and helped the All Blacks cling on to victory.
"I just had to stand up for the brothers," said Savea.
"For me I just try to anticipate things and do what the team needs me to do. It's as simple as that."
The teams meet a second time, in Wellington next week.