At the 59th minute, Ramires stole the ball and ran down the center before passing to Robinho, who fired past Bravo to put the Brazilians 3-0 up.
Slovakia's spirited campaign came to an end at Durban's Moses Mabhida Stadium when Arjen Robben, capping his first start in South Africa, helped send the Oranje into the last eight with an opening goal.
The World Cup debutants Slovaks will leave for home with their heads held high while the Dutch are now focusing on Friday's clash with Brazil in Port Elizabeth.
Vladimir Weiss's team started with a warning shot at the heavily fancied team as Erik Jendrisek blasted over the bar only two minutes into action.
The Dutch certainly didn't shudder as Robin van Persie and Wesley Sneijder both had good chances to score before the Bayern Munich winger put the team ahead.
Running down on a Sneijder's pass, Robben shook off Radoslav Zabavnik, cut inside and fired a low drive from the near post, beating goalkeeper Jan Mucha.
Slovakia had chances for an equalizer just after the hour mark as Dutch-based striker Miroslav Stoch forced a tip-over save from Maarten Stekelenburg. Then Robert Vittek missed a superb chance when he picked up the ball in an unmarked zone and shot directly into the palms of the Dutch keeper.
Even though Robert Vittek converted a penalty for the Slovaks in the dying seconds of stoppage time, which was awarded after Stekelenburg had brought him down, it was too late.
The Slovakian coach was gracious after the defeat.
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