Tomas Holes and Patrik Schick scored in a 2-0 win in the opener on Sunday, meaning the Czechs will face Denmark in the last eight after their Wales 4-0 battle on Saturday.
And it was a superb solo goal from Thorgan Hazard that proved decisive for Belgium, while they were beating Portugal 1-0 to condemn Cristiano Ronaldo to an early exit from what was a record fifth appearance in the European Championship finals for the Juventus star.
The Czech Republic were helped by the Netherlands who rose to 10 men in the 55th minute in Budapest, when Matthijs de Ligt was sent off for handball.
Holes put their country ahead with a powerful header in the 68th minute – and Schick wrapped up the win with his fourth goal of the tournament with 10 minutes remaining.
The Czechs, who only qualified in English Group D in third place, will now face Denmark next Saturday.
Belgium might consider themselves lucky to join them in the quarter-finals, given that Hazard’s 25-yard deflection effort in the 42nd minute was his first shot on target.
Portugal dominated possession and produced a grandstand finish after rallying in the second half, with their most agonizing chance coming when Raphael Guerreiro hit the post.
Belgium, who lost Manchester City talisman Kevin De Bruyne to injury, have survived and face Italy on Friday.
Meanwhile, UEFA has denied banning the rainbow flags from the Budapest fan zone and stadium following reports that Dutch and Czech fans confiscated them by security personnel.
UEFA insists it’s not because it banned the internationally recognized symbol of LGBT pride from venues, adding that it had asked the Hungarian Football Association to allow the installation of rainbow flags.
A statement read: “UEFA informed the Hungarian Football Association earlier today that the rainbow-colored symbols were not political and that, in accordance with the Equal Game campaign of the UEFA, which aims to combat all types of discrimination, including against the LGBTQI + community, these flags will be allowed in the stadium.
“Contrary to some reports in the Dutch media, UEFA would like to point out that it has not banned any rainbow symbols from the Budapest fan zone and that the fan zone is under the responsibility of the local authorities. “
it comes after UEFA controversially rejects request from Munich mayor to have the Allianz Arena illuminated in rainbow colors before last Wednesday’s match between Germany and Hungary.
The proposal was made days after Hungary’s parliament passed a new law banning the promotion of homosexuality or sex reassignment to those under 18.
UEFA said that although the rainbow symbol itself was apolitical, the request was political and was therefore denied.