The turning point of the game came in the 55th minute when Netherlands central defender Matthijs de Ligt was sent off for a handball when under pressure from Schick. The red card was given following a video review.
Midfielder Tomas Holes then put the Czechs ahead with a powerful close-range header in the 68th minute following a free kick from the right. He sprinted into the left of the penalty area and squared the ball to set up Schick for his goal in the 80th.
“The Dutch were the favorites and to beat them feels like unreal. It’s the best game of my life,” Holes said. “It’s the Euros, we had a lot of fans here and I really enjoyed it.”
Schick scored both goals in the team’s opening 2-0 victory over Scotland, then scored in the 1-1 draw with Croatia. Only Portugal forward Cristiano Ronaldo has more goals at Euro 2020 with five.
The Czechs will next face Denmark in the quarterfinals on Saturday in Baku, Azerbaijan.
The Czechs have reached the last eight of a major tournament for the first time since Euro 2012, when they lost to Portugal. Ronaldo scored the winner that day. They have not qualified for the past three World Cups and finished last in their group at Euro 2016.
On a warm evening at the 67,215-capacity Puskas Arena in Budapest, the orange-clad Dutch team paid for not creating enough chances. Then the Czechs made them pay even more.
With Schick practically on his back inside the penalty area, De Ligt slipped and scooped the ball away with his right hand. Referee Sergei Karasev awarded a free kick and, after jogging over to check his TV monitor, sent off De Ligt with a red card.
The incident came only moments after the Dutch could have taken the lead when forward Donyell Malen went clean through. He failed to get around goalkeeper Tomas Vaclik with an empty net waiting.
“Details change games and in one minute your world is turned upside down,” Netherland coach Frank de Boer said through a translator. “We had a chance for 1-0. These are hard lessons to learn.”
When De Ligt was sent off, de Boer took Malen off in a surprising move which summed up a poor performance for a team which had won all three of its group games.
“I’m responsible at the end of the day, so you can always blame it on me,” de Boer said. “I need to look in the mirror.”
Forward Memphis Depay and midfielder Georginio Wijnaldum combined for five goals in those three group matches but they had almost no impact on Sunday as the favored Dutch failed to even hit the target against the Czechs.
“Our players were not good enough,” de Boer said. “They didn’t play at their usual level.”
The referee used another video review in the 60th minute to rule out a penalty for a handball from Netherlands defender Denzel Dumfries.