Substitute Pilar was on hand to snatch all three points in time added on in Tuesday's Group A opener at Generali Arena to punish a calamitous mistake from defender Daryl Janmaat.
Newcastle United full-back Janmaat attempted to head back to Jasper Cillessen, but beat the goalkeeper and Pilar had the simple task of slotting home from point-blank range after the ball had struck the post.
There was no time for Netherlands to respond, so Guus Hiddink suffered a defeat in his first competitive game at the helm since he returned for a second spell as coach of his country.
Pavel Vrba, on the other hand, was able to celebrate his first victory since taking over as coach of the Czech Republic at the start of the year in what was his maiden competitive match in charge.
Borek Dockal had given the Czechs a half-time lead with a stunning left-footed strike, then Stefan de Vrij looked to have secured a point with a towering header for the visitors until Pilar had the final say in a dramatic late twist.
Demetris Christofi struck twice as Cyprus pulled off a major shock in Zenica, beating Bosnia-Herzegovina 2-1 in Group B.
The side ranked 140th in the world rallied following an early goal from Vedad Ibisevic, with Sion striker Christofi equalising on the stroke of half-time before doubling his tally to ensure Cyprus were able to savour one of their finest hours.
Gareth Bale came to Wales' rescue with a double to spare Chris Coleman's sides's blushes as Andorra were beaten 2-1 in Group B, having led through an sixth-minute penalty from Ildefons Lima - their first goal in a competitive match for four years.
There was concern for Aaron Ramsey, though, as the Arsenal midfielder was forced off with an injury late on.
There was another surprise in the Group A clash in Reykjavik as Iceland secured a 3-0 victory over Turkey.
The Nordic nation have never qualified for a major tournament, but made a great start to their European Championship qualifying campaign as goals from Jon Dadi Bodvarsson, Gylfi Sigurdsson and Kolbeinn Sigthorsson secured all three points.
Turkey also had Omer Toprak sent off for two bookable offences after 59 minutes as they endured a night to forget, while Kazakhstan and Latvia played out a goalless draw in the other game in that group.
Goals from Simone Zaza and Leonardo Bonucci ensured Antonio Conte marked his first competitive game as Italy coach with a 2-0 Group H win over Norway in Oslo.
Italy's victory came at a cost, though, with Borussia Dortmund newcomer Ciro Immobile carried off on a stretcher late in the game.
Conte has made a great start to coaching at international level, having also masterminded a 2-0 win over Netherlands in a friendly - which was his first match in charge last Thursday.
Croatia also began their bid to qualify from Group H with a 2-0 victory over Malta thanks to goals from Luka Modric and Andrej Kramaic after the visitors had Steve Brog sent off in the first half in Zagreb, while Bulgaria were 2-1 winners Azerbaijan courtesy of Ventsislav Hristov's goal three minutes from time.
