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Netherlands faced an uphill battle after Matthijs de Ligt was shown a straight red card early in the second half for denying a goalscoring opportunity.
The Czechs progress into the last eight of the competition for the first time since Euro 2012.
"Of course, it feels bad," an emotional De Ligt told NOS TV after the game. "We basically lost the match because of what I did.
"In hindsight, I shouldn't have let the ball bounce. We were in control before that and I feel responsible."
It was a miserable end for the Netherlands who were taking part in their first tournament knockout match since the 2014 World Cup in Brazil.
It was a slow start to the game with the Czechs having the first chance after 23 minutes when Tomas Soucek's diving header went past the post.
The Czechs had an even better chance after 34 minutes but De Ligt made a stunning block to deny Lukas Masopust from close range.
Netherlands found themselves down to 10 men moments later after a long VAR check adjudged De Ligt to have denied Schick a goalscoring opportunity with a handball outside of the box.
It was the first time a Netherlands player had been shown a red card at the Euros since John Heitinga against the Czechs in 2004.
The Czechs took the lead after 68 minutes when Tomas Kalas got on the end of free kick to set up Holes to score at the back post.
With 10 minutes remaining, the Czechs produced a counter attack and Holes set up Schick to convert from inside the box.
"They were the favourites in this game, it is an unreal feeling to beat such a team 2-0," Holes said.
"They were better with the ball, we struggled, but thanks to our team performance, we did not give them much space."