The Dutch needed to beat the already-qualified Czech Republic in Amsterdam and hope Turkey lost at home to already-qualified Iceland in Konya to salvage third place in Group A and a spot in next month's play-offs.
But in the end neither happened, with the 1988 European champions going down 3-2 to the Czechs, their third loss in four qualifiers under Danny Blind, while Turkey beat Iceland 1-0.
Pavel Kaderabek of Hoffenheim put the Czechs ahead and Josef Sural doubled their lead, before the away side had Marek Suchy sent off for chopping down Memphis Depay as he raced through on goal two minutes prior to half-time.
Klaas-Jan Huntelaar and Van Persie scored at the other end to reduce the deficit but by that point the damage had been done and the Dutch will miss a major tournament finals for the first time since the 2002 World Cup.
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They have not failed to qualify for a European Championship since 1984, when the finals were also played in France.
"I have no intention to step down. I'm going to carry on with my work because I'm under contract until 2018 and I believe in this team," Blind, who replaced Guus Hiddink in July, said afterwards.
Turkey, meanwhile, had Gokhan Tore sent off 12 minutes from time against Iceland but had third place in the bag thanks to events in Amsterdam.
They went one better, though, as Selcuk Inan's brilliant free-kick on 89 minutes saw them win their game. And that, coupled with a 1-0 win for Kazakhstan in Latvia -- secured by an Islambek Kuat goal -- allowed Fatih Terim's side to progress automatically as the best third-placed side.