Stand-in TJ Perenara took full advantage of his opportunity at a packed Kings Park Stadium to score two tries in an ultimately emphatic win born out of relentless pressure that delivered seven second-half tries.
New Zealand, who now share the record with South Africa, will become the first top-tier team to win 18 Tests in a row if they defeat Australia on October 22 in Auckland.
New Zealand fly-half Beauden Barrett, who began the game as the leading Rugby Championship scorer this season with 65 points, fluffed his first penalty attempt.
Africa were gaining little possession but defended with a lot more spirit than on a recent, disastrous tour in which they lost to Argentina, Australia and New Zealand.
Steyn slotted another penalty to double the lead to six points before the All Blacks scored their first try through right winger Israel Dagg on 22 minutes.
It was a typical New Zealand score with backs and forwards applying pressure in the danger zone and when the ball was moved wide, a quick pass from flanker Jerome Kaino sent Dagg over.
Barrett was out of luck with the conversion, which struck the near post, leaving the Springboks 6-5 ahead.
The visitors took the lead for the first time when Perenara was awarded a try after numerous replays were shown to the referee, his assistant and the television match official.