The 51st minute spot kick looked to be a harsh decision against Gregory van der Wiel, who got to the ball with his tackle as Birkir Bjarnason tumbled over his feet.
But Sigurdsson squeezed the penalty under Dutch goalkeeper Jasper Cillessen to take the North Atlantic island nation to the brink of a first major championship appearance.
Iceland completed a double over the Dutch in Group A and moved to 18 points, their lead in the standings staying at two over the second-placed Czech Republic who came from behind to beat Kazakhstan 2-1.
(Reporting by Mark Gleeson, editing by Mike Collett)
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