The former European champions had to win and hope Turkey lost at home to Iceland to sneak into third place in Group A and book a slot in next month's playoffs.
However, the Netherlands were consigned to fourth spot behind the Turks, who won 1-0 to secure a place at the finals in France as the best third-placed team.
The loss left the Dutch with 13 points from their 10 matches behind Turkey on 18, while the Czechs claimed top spot in the group with 22 points, two ahead of Iceland.
Pavel Kaderabek and Josef Sural stunned the hosts with goals in the first period and the game was over when substitute Robin van Persie headed into his own net in the 66th minute. The Dutch consolation goals came from Klaas-Jan Huntelaar and Van Persie.
"The truth is we gave it away," said midfielder Wesley Sneijder told Dutch NOS radio. "We needed to be disciplined and sharp because we weren't going to get too many chances.
"We knew it would be tough but to so easily concede goals meant it was all over for us early in the game."
A second half change in tactics by the Dutch to full-out attack failed to turn around a dire situation despite the Czechs being down to 10 men after the sending off of Marek Suchy just before halftime for a last-man tackle.
By then they were 2-0 down with the Czechs having taken the lead midway through the first half with some simplicity, exposing some shoddy defending as the Netherlands went behind for the fourth time in five home matches in the group.
Fullback Pavel Kaderabek was allowed to run in from the left after some clever passes from Tomas Necid and Jiri Skalak and slam the ball home with ease after 24 minutes.
Eleven minutes later they scored again courtesy of more woeful Dutch defensive work. Virgil van Dijk turned his back on a tackle and allowed Josef Sural to force his way down on goal and beat goalkeeper Jeroen Zoet at his near post.
IMMEDIATE CHANGE
It prompted an immediate change from Netherlands coach Danny Blind with striker Robin Van Persie brought on and defender Jairo Riedewald sacrificed.
The home side were given a glimmer of hope just three minutes before halftime when Memphis Depay was caught by a mistimed tackle from Suchy, who was sent off.
But despite even more attacking changes after the break, the hoped for comeback did not materialise as the Netherlands hoofed long balls towards the Czech defence.
The Dutch woes were complete midway through the second half when Van Persie, in his 101st appearance, headed an innocuous looking free kick from Jiri Skalak into his own net.
A largely silent crowd finally found something to cheer with 20 minutes to go when Huntelaar rose above the defence to head past goalkeeper Petr Cech.
Van Persie then scored at the right end as he poked home a flick on from Bas Dost with seven minutes left.
The Netherlands were European champions in 1988 and third in 1992, 2000 and 2004. They went out at the group stage of the last edition at Euro 2012 in Poland and Ukraine.
(Reporting by Mark Gleeson; Editing by Ken Ferris)
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