It was the World Cup semi-finalists who started the strongest, but chances were difficult to come by as the Czech defence held firm for the majority of the first half.
The deadlock was broken by the Czechs in the 22nd minute when Borek Dockal received the ball from David Lafata on the edge of the box, before firing an unstoppable half-volley home off the inside of the post.
Guus Hiddink's side wasted a chance to draw level almost immediately as Georginio Wijnaldum was caught in two minds while in plenty of space after Daley Blind had picked out his teammate with an inch-perfect cross.
Improvements were made by the visitors when Hiddink abandoned the 3-5-2 formation late in the first half, but their frustration continued as Petr Cech pushed over a powerful effort from Memphis Depay.
The Dutch started the second half in control, and it was another Blind delivery which produced the equaliser in the 56th minute as a deep cross reached Stefan de Vrij, whose header left Cech with no chance of protecting his side's lead.
Chances were rare following the equaliser, but Vladimir Darida did come close to restoring the lead for the hosts when his low shot from the edge of the box drifted just wide of the post.
The pressure was on the Czech defenders as the match entered the closing stages, and they were left breathing a sigh of relief when Cech kept hold of a Depay header.
It should have been 2-1 to the hosts late on, but Vaclav Prochazka missed a chance when he headed over at the back post from Tomas Rosciky's corner.
However, there was still time for a late twist as Janmaat inadvertently headed against his own post while trying to find Jasper Cillessen, allowing Pilar to tap home from close range to secure the win.