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Despite the Netherlands' dominating for much of the first half, Iceland managed to double their lead before half time when Kolbeinn Sigthorsson had a shot blocked, setting up Sigurdsson for a volley on the rebound.
Bruno Martins Indi and Robben both tried to get the visitors into a goal-scoring position early in the second half, but they were unable to find their teammates ahead of them.
Quincy Promes came off the bench, and almost set up Van Persie to get a goal back, but the Manchester United striker headed wide.
Iceland almost added a third late on when Jon Dadi Bodvarsson played his way into the box before blasting the ball towards the Dutch goal, only to watch his effort go wide.
The result leaves Iceland with maximum points from their first three qualifiers at the top of Group A, while the Netherlands sit third.