Robinho, who had already had an effort ruled out for offside, fired the Brazilians ahead with just 10 minutes gone, and the Dutch struggled to cope during the opening 45 minutes. But Felipe Melo's 53rd-minute own goal gave them fresh impetus and Sneijder won it 22 minutes from time as Melo was dismissed for stamping on Arjen Robben.
However, Holland will have to do without both full-back Gregory van der Wiel and midfielder Nigel de Jong for Tuesday's semi-final in Cape Town after both collected second bookings. Dutch legend Johan Cruyff had claimed dismissively before the game that he would not pay to watch this Brazil team, and if he did stay away from the Nelson Mandela Bay Stadium, he may have been glad he did so at half-time.
- Netherlands v Brazil: As it happened
Dunga's men may not possess the all flair of their forebears, but they had more than enough to cut Holland open almost at will, and the 1-0 lead they enjoyed at half-time was scant reward for their dominance. Robinho thought he had given the South Americans the lead with just eight minutes gone after Luis Fabiano and Dani Alves combined to carve open the path to goal and he supplied the finish.
An offside flag ended that celebration, but the #32.5million man did not have to wait too long for a chance to ease his disappointment. This time, it was defensive midfielder Melo who split the Dutch rearguard wide open with a straight ball down the middle, and Robinho left orange shirts floundering in his wake as he beat keeper Maarten Stekelenburg with consummate ease.
But for Stekelenburg, the game would effectively have been over by the break with the Ajax man pulling off a fine one-handed save to keep out Kaka's curling 31st-minute effort after Robinho and Luis Fabiano had left Holland once again chasing shadows down the left.
The keeper was relieved to see full-back Maicon's rasping drive, which bore similarities to Carlos Alberto's stunning strike in 1970, tear into the side-netting in injury time with the Dutch very much up against it.
But they resumed in more determined fashion, and they got their reward within eight minutes of he restart when, after Sneijder had curled a cross hopefully into the penalty area, Julio Cesar and Melo challenged each other and the ball went in off the midfielder's head. Robben started to cause problems in the wake of the goal, but Alves flashed a long-range effort just wide and Kaka also went close at the other end as Brazil responded.
But it was Holland who took the lead with 22 minutes remaining when Kuyt flicked on Robben's corner and Sneijder steered a header into the net. Melo received his marching orders five minutes later for senselessly stamping on Robben with his side's World Cup dream rapidly unravelling. Andre Ooijer's last-ditch tackle denied Kaka five minutes from time, and there was no way back for the five-time winners.