But the 2019 runners-up suffered some anxious moments in front of 40,000 in Sydney and had goalkeeper Daphne van Domselaar to thank for keeping the African champions at bay in the first half.
Jill Roord scored after nine minutes to put the Dutch on their way against a team ranked 45 places below them and featuring in the knockout rounds for the first time.
South Africa goalkeeper Kaylin Swart then made a mess of an innocuous Lineth Beerensteyn shot midway through the second half to give the Dutch a second.
The Netherlands will meet Spain on Friday in the last eight but Van Domselaar was called into action numerous times.
“It wasn’t our best game but happy to play my part in the clean sheet,” said the stopper, who plays for Aston Villa and was named player of the match.
“I don’t know if it was my best (game) but I will be looking back with pride.”
Netherlands coach Andries Jonker was relieved afterwards that his side avoided becoming the latest to be on the end of a World Cup shock.
He said his team can beat anyone, but was scathing of their first-half performance.
“We lost the ball too often in the first half and gave a few chances away,” he said. “Daphne saved us a few times and I was happy to lead 1-0 at half-time. We did much better in the second half and had much more control of the game. We deserved to score again and in the end deserved to win.”
The 54th-ranked South Africa will be kicking themselves after having numerous chances in the opening 45 minutes, almost all to their livewire captain Thembi Kgatlana.
On six minutes the striker fired straight at Van Domselaar — it was to become a theme.
The Dutch took the lead three minutes later, midfielder Roord nodding in from close range following a corner for her fourth goal of the tournament.
Kgatlana, who scored the winner over Italy that gave South Africa their first win ever at the World Cup, was a constant threat with her pace.
Four more times in the first half the US-based Kgatlana could have scored.
South Africa coach Desiree Ellis was proud but felt her side should have won.
“We had a lot of chances, especially in the first half, to kill off the game,” she said. “The chances we created should have put us out of sight.”
Neutrals at Sydney Football Stadium were behind South Africa and there was a buzz of anticipation each time Kgatlana ran with the ball.
On 12 minutes she sprinted away from a Dutch defence that was treading water and fizzed her effort just over the bar.
Kgatlana rattled the Dutch once more soon after with a mazy run and shot that stung the palms of Van Domselaar.
South Africa had already taken off injured forward Jermaine Seoposenwe and they lost Bambanani Mbane three minutes before the break when she was stretchered off with a badly hurt ankle.
If anything that spurred Ellis’s side and Kgatlana surged through again but was foiled by Van Domselaar once more.
Moments before the break there was a repeat, the Dutch defence finding the 27-year-old Kgatlana too hot to handle but Van Domselaar standing firm.
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The Dutch got a grip on proceedings in the second half and were gifted a second on 68 minutes when Swart’s blunder let Beerensteyn’s tame effort slip through her hands.
“In the end the legs were not there to push for the duration of the game,” said Ellis.
She added: “Their goalkeeper getting player of the match shows how well we played. The whole of South Africa should be very proud.”