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 guarantee their place at Euro 2016 with a home win over Kazakhstan on Sunday.
Holland also had Bruno Martins Indi sent off in the first half after clashing with Kolbeinn Sigthorsson.
<#Euro2016>