Five-times champions Brazil looked firmly in control in the first half, Robinho giving them a lead in only the 10th minute, two minutes after having a goal ruled offside.
Wesley Sneijder headed a second-half winner as the Netherlands battled back from a goal down to beat ten-man Brazil 2-1 on Friday and secure a place in the World Cup semi-finals.
Brazil, five times champions, took a tenth minute lead through Robinho and dominated the first half against a Dutch team which initially looked to have been outclassed.
But the Dutch levelled out of the blue in the 53rd minute after a rare mistake from Brazil's usually impenetrable defence, as Felipe Melo headed Sneijder's cross into his own goal.
Then Sneijder headed a winner as the Dutch, battling to their 13th successive World Cup win to maintain a perfect record in the qualifiers and finals, avenged a quarter-final defeat in 1994 and a semi-final defeat four years later.
"In the changing room at half-time, everyone said to each other, 'let's give everything', and that's what we did. We fought for each other," Sneijder told a television reporter after the match.
"In the second half, we put a lot of pressure on their defence and to score twice was fantastic."
He added: "We're in the last four of the world. To beat Brazil 2-1 in the quarter-final, you have to be pleased with that. But in the next days, we will have to get our heads in focus again."
Dutch coach Bert van Marwijk said the comeback showed how mentally strong his team were.
"In the first 20 minutes, I was happy that the score was only 1-0, and I told Frank [assistant Frank de Boer] that we still have a chance.
"In the second half we showed how good we are.
"Tonight we're going to party and then start and fight again."
Brazil were left shocked by the defeat.
"We are all extremely sad and we didn't expect it. We were not able to maintain the same rhythm in the second half," Brazilian coach Dunga told a post-match news conference.
"It's not the end of the world," goalkeeper Julio Cesar told a TV reporter as a tear rolled down his face.
"I just think now we have to leave with our heads held high. In Brazil people always put a lot of pressure on us. This was a result no one expected. Every game Brazil enters, we enter to win."
Brazil had started brightly with Felipe Melo, returning from an ankle injury which kept him out of the second round match against Chile, setting up the first goal after 10 minutes of the clash at the Nelson Mandela Bay Stadium.
He threaded a 30-metre pass through the Dutch midfield and Robinho timed his run perfectly to score with a first-time shot past Maarten Stekelenburg.
The game turned niggly, although Brazil managed to produced some flowing moves and should have extended their lead.
Juan volleyed over from close range after a jinking run on the right by Dani Alves and Stekelenburg produced a superb one-handed save to tip away Kaka's curling effort, following a dazzling Robinho run down the left.
Just before half-time, Maicon fired into the side netting after Luis Fabiano rolled the ball into his path.
The Dutch rarely threatened with Arjen Robben far too predictable to outfox the Brazil rearguard.
But they levelled out of the blue eight minutes after the break when Felipe Melo touched Sneijder's looping cross-shot into his own net.
Kaka missed an excellent chance to put Brazil back ahead, side-footing wide from the edge of the area after the ball was gifted to him by a poor clearance.
They paid dearly in the 68th minute when Dirk Kuyt flicked on Arjen Robben's corner at the near post and Sneijder headed into the net.
Shortly afterwards, Felipe Melo was sent off for stamping on Robben.
"If you play with only 10 players against a quality team like Holland then it makes it difficult," Dunga said.
In the semi-finals, The Netherlands will meet the winners of Friday's second quarter-final at Soccer City in Johannesburg between Uruguay and Ghana.
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