The Czechs, without injured captain Tomas Rosicky, took the lead in the 72nd minute when Petr Jiracek collected a pass from Milan Baros and cut inside a defender before slotting past keeper Przemyslaw Tyton for his second goal of the tournament.
The Czechs advance with Greece, who upset Russia 1-0 in Warsaw, and will play the runners-up of Group B.
"We did what we could do. I don't know how to explain the fact that in one game boys played like they were fired up while in the second game they didn't. It is the end of this great adventure," said Poland coach Franciszek Smuda.
Captain Jakub Blaszczykowski added: "All in all we had great moments during this tournament and all we can say is 'thanks'. This is the kind of a moment, when one needs to say we need to move on, rethink everything, but this experience is essential."
The co-hosts, roared on by the crowd and needing a win to go through, started the game at a furious pace and will regret their profligacy after missing several chances to go ahead.
The Czechs were fortunate when a sloppy back pass left striker Robert Lewandowski with a chance but he skewed his effort off target under pressure from Theodor Gebre Selassie.
Clearly missing their playmaker Rosicky, the Czechs eventually settled down after a nervy opening 15 minutes and began putting the Polish defence under pressure with Vaclav Pilar providing a threat down the wing most of the night.
The Czechs started the second half strongly knowing they needed a goal to book a quarter-final ticket with Pilar causing all sorts of headaches with his runs at the Polish backline.
The pressure finally paid off with midfielder Jiracek's neat finish into the bottom corner of the net following a swift counter-attack after a pass from much-criticised forward Baros who played his best game of the tournament so far.
Poland frantically poured forward in search of an equaliser to keep their hopes alive but the Czechs defended in waves and held on to advance to the last eight.
