The Dutch needed to win and for Iceland to beat Turkey to earn a play-off spot, but Van Persie's error saw them go 3-0 down despite Mark Suchy seeing red.
Klaas-Jan Huntelaar and Van Persie replied to goals from Pavel Kaderabek and Josef Sural but instead it was Turkey who secured third place.
Fatih Terim's side beat Iceland 1-0. Netherlands coach Danny Blind took over at the start of July after Guus Hiddink left after 10 months in the position.
