The Czechs had a two-goal lead inside seven minutes thanks to Vaclav Pilar and Petr Jiracek, but a Theofanis Gekas strike gave the Greeks a way back into the match in the second half after a mistake from Cech in goal.
Cech failed to catch a simple ball just inside his own penalty area, with the 30-year-old colliding with Tomas Sivok, but thankfully for the Chelsea goalkeeper Greece were unable to build on that goal and Czech Republic held on for the win.
Speaking about the victory and his mistake to reporters, Cech said: "I must admit that I'm happy and felt relief that we have won. We had a fine start, the first 10 minutes were magnificent.
“After Greece's goal they pushed us but there were no chances only crosses and we played well defensively, handled the situation like we should.
“As for my mistake, I saw that Tomas [Sivok] tried to stick his leg to it so I was uncertain for a second and he was standing quite close. I wasn't making a complaint to him, it was just pure frustration at giving the ball away.”
The Czechs have one more match remaining in Group A with a crucial face-off with co-hosts Poland, but Cech admits there is a worry over the fitness of captain Tomas Rosicky.
“We will look at the Poland-Russia game obviously but for us the most important thing is whether Tomas Rosicky will play on Saturday.”