- Jill Roord (9')
- Lineth Beerensteyn (68')
Roord's header gave her side an early lead and Lineth Beerensteyn added the second goal courtesy of a goalkeeper error in the second half to secure the Dutch a date with Spain in Wellington on Friday.
The 54th-ranked African champions never gave up the fight in their first appearance in the World Cup knockout stage and were always a threat on the break through their lone striker Thembi Kgatlana.
Dutch goalkeeper Daphne van Domselaar was equal to everything Banyana Banyana fired at her, however, and the 2019 finalists progressed to the last eight for the second successive tournament.
The noon kickoff - primetime in the United States - was tailor-made for the reigning champion Americans but they were packed off to Melbourne to face Sweden later on Sunday after the Netherlands hammered Vietnam 7-0 to win Group E.
The Dutch started well and took the lead from a corner in the ninth minute when Danielle van de Donk headed the ball towards goal and defender Lebohang Ramalepe got in goalkeeper Kaylin Swart's way to present Roord with a simple finish.
Banyana Banyana responded immediately, striker Thembi Kgatlana's fine first touch on a through ball giving her space to unleash a shot that forced a save out of van Domselaar.
Kgatlana was a constant warning to the Dutch against complacency.
She got three shots away in the 10 minutes before halftime that tested van Domselaar, who also had to be at her best to stop a Kholosa Biyana piledriver in stoppage time.
Ten minutes into the second half, winger Lieke Martens had a goal disallowed for offside, but the Dutch did double the lead in the 68th minute.
Beerensteyn was played through and shot across Swart, who appeared to have the everything under control but fumbled her catch and watched the ball squirt into the net.
South Africa again responded to conceding a goal but van Domselaar was again equal to it, getting down low to turn Linda Motlhalo's low drive around the post five minutes later.
Banyana Banyana had one last chance to get on the scoreboard in stoppage time, almost inevitably through Kgatlana, but there was no way past van Domselaar.
- Yoshimi Yamashita