The referee awarded the spot kick in the 51st minute even though Gregory van der Wiel seemed to win the ball in a challenge with Birkir Bjarnason.
Sigurdsson squeezed the penalty under goalkeeper Jasper Cillessen to take Iceland to the brink of their first major championship finals.
It completed a fairytale double over the Dutch in Group A as Iceland took their points total to 18, two ahead of second-placed Czech Republic. The Dutch are third on 10 points.
Iceland can now book a place at Euro 2016 by scoring a home win over Kazakhstan on Sunday.
The home defeat provided a horror start for new coach Danny Blind. His predecessor Guus Hiddink quit because of the team's unconvincing performances earlier in the campaign, including the 2-0 defeat in Iceland in October.
Robin van Persie, who led the attack at the World Cup in Brazil just over a year ago, was left on the bench by Blind.
New captain Arjen Robben and midfielder Wesley Sneijder put together some lively combinations early on but the injury-prone winger went off after 27 minutes with a groin problem.
There were more Dutch woes six minutes later when Bruno Martins Indi lashed out at Iceland striker Kolbeinn Sigthorsson as both tumbled to the ground and the Dutch defender was sent off.
Blind's men created few chances until late on when goalkeeper Hannes Halldorsson made good stops from Georginio Wijnaldum and Sneijder. The visitors went close to a second goal when Johann Gudmundsson struck the upright with a curling drive after being teed up by Bjarnason.
Iceland took 3,000 fans to Amsterdam, around one percent of the small Nordic island nation's population.
(Editing by Tony Jimenez; mark.gleeson@thomsonreuters.com; +27828257807; Reuters Messaging: Reuters Messaging: mark.gleeson.thomsonreuters.com@reuters.net)
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