The Swansea City midfielder struck twice in the first half of Monday's Euro 2016 Group A qualifier in Reykjavik as Hiddink's men fell to a third defeat in four games since he retook the reins.
Iceland, who narrowly missed out on qualification for the recent World Cup, have established themselves as a force to be reckoned with in recent times.
And having opened their qualifying campaign with 3-0 victories over Turkey and Latvia, they secured one of their most famous wins in their history at Laugardalsvollur as Sigurdsson took his tally to four goals in his last three international appearances.
History was not on Iceland's side, given they had lost nine and drawn one of the previous two competitive meetings between the two nations.
But they ended their wait for a triumph over the Netherlands in spectacular fashion to ensure they top the group level on nine points with Czech Republic after another superb performance, while Hiddink's side have only three points from as many qualifying matches.
Both sides enjoyed the luxury of naming unchanged teams, having emerged triumphant in qualifying fixtures last week.
The pattern of the game was quickly established as Iceland looked to sit back and counter their illustrious opponents.
The hosts' ability to adopt such an approach was aided greatly when they were gifted a 10th-minute opener.
Gregory van der Wiel was the culprit as he needlessly brought down Birkir Bjarnason in the area, paving the way for Sigurdsson to dispatch a powerful penalty into the corner of the net.
The visitors continued to dominate possession, although clear-cut opportunities were not forthcoming.
Arjen Robben was at the heart of their finest work and he set up Robin Van Persie, but the striker was denied from a tight angle by the onrushing Hannes Halldorsson.
The visitors' best opening of the first half came soon after, Robben somehow heading wide after connecting with Daley Blind's pin-point cross.
And they were rueing that even more in the 41st minute, when Sigurdsson capitalised on the Netherlands' failure to clear a corner by smashing a stunning volley into the roof of the net.
Hiddink wasted little time in introducing Klaas-Jan Huntelaar, who sparked a turnaround from the bench in the 3-1 win over Kazahkstan on Friday, at half-time in place of Wesley Sneijder.
Yet it was the home side who continued to look the most threatening, Bruno Martins Indi's brilliant last-ditch clearance turning away Birkir Mar Saevarsson's cross with Kolbeinn Sigthorsson lurking at the back post.
The Netherlands were still enjoying plenty of possession but, despite continuous probing, could not craft any real openings against a well organised Iceland side.
Iceland remained in control and could even have secured a third successive 3-0 success, when Jon Dadi Bodvarsson slid an effort narrowly wide but they celebrated on the pitch at full-time after claiming a major scalp.
