Sestak and strike partner Robert Vittek were causing the Northern Ireland defence problems while Weiss was a constant danger on the right.
Slovakia were on course for a perfect start in their debut World Cup as an independent nation after Robert Vittek's 50th-minute header.
Seconds later Marek Hamsik toe-poked forward for Vittek, who showed none of the composure he demonstrated to score twice against Italy by firing straight at Stekelenburg.
Capello will have been concerned, however, by defensive frailties that meant England keeper David James had to produce two fine first-half saves from Miroslav Karhan and Robert Vittek.
James was in action again after 40 minutes, turning a shot from Vittek over the top after a mix-up between Ashley Cole and Matthew Upson had let in Slovakia's captain.
Slovakia finally found a good delivery from Sestak on the right, the Bochum winger whipping a cross in from deep for Vittek to plant a perfect header into the bottom corner.
Slovakia's final touch in this World Cup was a positive one as Vittek stroked home a penalty for his fourth and final goal of the tournament after Stekelenburg had tripped the striker.
Sneijder stroked home a pull-back from Liverpool's Dirk Kuyt to essentially seal the game for the Dutch, with Robert Vittek firing home a consolation penalty with the very last kick of the match.
The Wellington Phoenix player embarrassingly miscued a clearance, gifting Vittek a difficult chance, and was fortunate to see one of his defenders clear when he failed to collect a header across the box.
Their attitude was summed up when more good work from Sestak gave Vittek a chance to wrap it up, but the striker appeared to delay too long, allowing Reid to slide in with an excellent tackle.
In Group B , French side Lille moved two points clear at the top with a 3-0 crushing of Italy's Genoa, with Ludovic Obraniak scoring seven minutes before halftime and Robert Vittek and Eden Hazard adding goals after the break.
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