Sigurdsson had been a doubt with an injury sustained against Latvia on Friday but the Swansea City midfielder featured from the start and was worth his weight in gold for the hosts.
He scored his first goal in the 10th minute of the Group A game when he converted a spot kick conceded by Stefan de Vrij, who fouled Birkir Bjarnason inside the penalty area.
Sigurdsson then doubled his tally three minutes before halftime when he fired home after being quickest to react to a loose ball after the Dutch failed to clear a corner.
The Dutch, who remain on three points, face a tough task to progress as they trail group leaders Iceland and Czech Republic, who earlier beat Kazakhstan 4-2, by six points in the table.
The Netherlands had never lost to Iceland, having won nine out of 10 previous matches -- the other was a 1-1 draw in Reykjavik in 1982 -- but could not break down their defence.
The hosts were full of confidence after 3-0 victories over Turkey and Latvia in their previous qualifiers.
Netherlands coach Guus Hiddink threw Klaas-Jan Huntelaar for Wesley Sneijder at halftime but, unlike against Kazakhstan in their last qualifier when the Shcalke 04 striker scored, he could not get the Dutch out of their plight this time.
(Writing by Geert Beckers, Editing by Ken Ferris)
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