Robben scored a trademark goal in the 18th minute in what turned out to be a cruise for the Dutch, who did what was required without over-exerting themselves, before Sneijder sealed the success while Robert Vittek grabbed an injury-time consolation for his nation.
The triumph means Holland will next face either Brazil or Chile in the last eight, with the South American pair to play on Monday evening.
Robben handed the Dutch a boost by making his first start of the 2010 World Cup, replacing Rafael van der Vaart, while Gregory van der Wiel returned to the team. For Slovakia, head coach Vladimir Weiss selected his son of the same name, with the Manchester City youngster coming in for Zdeno Strba.
Slovakia enjoyed a confident start to the fixture as a bright counter-attack in the second minute saw Weiss pass to Erik Jendrisek, whose strike from outside the box zoomed over the bar. Then it was Holland's turn for a sighter, Sneijder the man to lash off target.
In the sixth minute, Marek Hamsik's stinging left-footed shot had Maarten Stekelenburg a little more concerned, but the ball whistled wide. A minute later, Robin van Persie was presented with the game's best chance thus far when he met a Dirk Kuyt cross, the Liverpool man delivering from the left wing, only to see his tame header deflected out for a corner.
Holland struggled to find tempo and rhythm during the early exchanges, although a rare flash of flair from Van Persie, as he turned well before slipping in Sneijder, opened up the Slovaks. But the Inter Milan schemer's effort was rather scuffed resulting in an easy stop for goalkeeper Jan Mucha.
However, Holland broke the deadlock on 18 minutes. A fabulous raking pass from Sneijder deep inside his own half was latched onto by Robben, who typically cut inside to open up his body for a left-footed strike, with Van Persie's run helping to create space, and the Bayern Munich man rifled low into the corner.
Slovakia struggle to respond to their setback as the Dutch looked comfortable without being exerted. Indeed, Slovakia's next most half-threatening moment came in the 31st minute when Nigel de Jong cleared a deep free-kick in unorthodox fashion. Seconds after, Robben was perhaps harshly given a yellow card for a deliberate handball.
The first half ended with a whimper as a lack of invention from both sides proved the undoing of a spectacle that promised much at the start. Moments of note before the half-time whistle included another harsh booking, this time for a Juraj Kucka foul, and a tepid right-footed shot by Van Persie. But that is scraping the barrel.
Holland very nearly doubled their lead twice in the space of a minute early in the second half. First, another Robben effort was expertly pushed around the post by Mucha before the Slovak keeper was swiftly called back into action, this time denying Joris Mathijsen with his face after a cross by Robben.
Van Persie then found himself involved on two differing occasions when he caught defender Radoslav Zabavnik with an awkward challenge before the Arsenal striker sent in an in-swinging free-kick which was bravely punched away by the Everton-bound Mucha.
Finally, Stekelenburg was called into action as twice in a minute Slovakia hit the target. Miroslav Stoch was first to pull the trigger, forcing a palm over from Stekelenburg before an even better chance for Vittek, who was one-v-one with the keeper, drew a stop which was made simple for the fact that the strike was central.
Holland boss Bert van Marwijk replaced the not yet fully fit Robben with Elijero Elia on 71 minutes while Jendrisek was replaced by Kamil Kopunek, whose first act was to foul Elia and be booked for the challenge.
Kuyt was the man to almost double Holland's advantage when he headed over then thumped a long-range struck that stung the hands of Mucha. Vittek was presented with a half-chance with 12 minutes remaining but blasted over before Klaas Jan Huntelaar entered the fray for the rusty Van Persie.
And with six minutes left, the result was put beyond doubt. Mucha, so excellent throughout the game, was at fault as he came racing out of his goal to allow Kuyt to nip in. The Liverpool forward got his head up and rolled across goal for Sneijder who buried into the exposed goal and ensured of Holland's progress.
But there was to be some cheer for Slovakia when Vittek grabbed his fourth goal of the tournament in the fourth minute of injury-time, dispatching from the penalty spot after a foul by Stekelenburg on Martin Jakubko in the area. Yet it proved nothing more than a consolation.Material: skysports.com