The Black Ferns are the record six-time Women's World Cup winners and have not lost a match at the tournament since 2014.
By contrast South Africa, unlike their four-time world champion men's team, were appearing in their first quarter-final.
Yet at half-time the Springboks were level at 10-10 and threatening a huge - and rare - upset at a World Cup dominated by lopsided wins for leading teams.
But New Zealand then scored 29 unanswered points to put the result beyond doubt and set up a semi-final with either Canada or Australia, who play later on Saturday.
The Springboks stunned the Black Ferns by opening the scoring in Exeter through prop Babalwa Latsha.
New Zealand hit back through tries from centre Theresa Setefano and teenage wing Braxton Sorensen-McGee, but Aphiwe Ngwevu's try levelled the game at half-time to round off a superb opening 40 minutes for South Africa.
But the Black Ferns regained the initiative after the break with five tries. Full-back Renee Holmes scored twice, with Sorensen-McGee, the tournament's top try-scorer, and No 8 Kaipo Olsen-Baker also completing try doubles.
South Africa managed to finally cross late in the second half through Lerato Makua to finish off a memorable campaign on a high in what is set to be veteran captain Nolusindiso Booi's final game after the 40-year-old lock said she would retire when the Springboks were no longer involved at this World Cup.
New Zealand replacement Katelyn Vahaakolo was the final try-scorer of an entertaining clash.