As for Fernando Santos's side, they have key issues to address before a daunting final group game against Russia. Santos has already used 17 of his squad in the opening two games and you wouldn't rule out seeing some of the remaining six after this. We look at the issues that both teams need to think about after this result.
Greece looked like being overrun against Poland in the opening game before hauling themselves back into it despite having been reduced to 10 men. Here they were two goals down before they'd even started playing and again managed to make a game of it in the end. It's not a new phenomenon. Although they were unbeaten in qualifying, the Greeks were actually losing more matches than they were winning at half-time. Is the problem a tactical one or a mental issue? Perhaps it's even a credit to their stamina? Whatever the reason, it is something they surely need to get to the bottom of.
Santos has been struggling for some time to settle on his first-choice goalkeeper. Orestis Karnezis played against Belgium in February but then didn't even make the squad. Dionisis Chiotis was called up in October and appeared to have cemented his place after performing heroics in APOEL's run to the quarter-finals of the UEFA Champions League. And yet, he too was omitted. Ultimately, Santos settled on 38-year-old Kostas Chalkias but the veteran was unable to reward his coach's faith and was partly culpable for both Czech goals in this one before trudging off injured with the game still in the first half. A regret for Santos, no doubt.
The Greece left-back endured a tough examination in the opening game against Poland where he found himself overrun by Jakub Blaszczykowski and Lukasz Piszczek at times - a problem he contributed to by taking up an advanced starting position. If there was any hope that Holebas had taken something positive from that experience it was dashed within moments of this game kicking off. The 27-year-old was far too advanced as the Czechs attacked for the first time and he could only saunter back as Jiracek powered forward to score. Holebas is a force going forward but that will be forgotten as long as he continues to undermine his team's chances defensively.
Theodor Gebre-Selassie certainly appeared to have watched Piszczek's first-half performance against Greece. The right-back had been one of the few bright sparks going forward in the 4-1 defeat to Russia and again he showed his ability to get forward. In this instance, he was exposing a Greek weakness and it was no coincidence when one such driving run was rewarded for the Czech Republic's second goal.
Need to win
Although the Czech Republic were in the unenviable position of knowing they would be the first team eliminated from the tournament if they lost this game, that appeared to give Bilek's side the certainty they needed early on. While Greece's point against Poland gave them the advantage on paper it seemed to be a disadvantage on the pitch. They started like a side that had done the maths and were busy calculating what they needed. The Czechs, free from doubts, went out and seized the initiative.
The stadium in Wroclaw was a sea of red as the Czech Republic brought good support from over the border. They were in good voice before kick-off despite the misgivings they must have had after the spanking at the hands of Russia on Friday. Their enthusiasm seemed to seep down onto the pitch and the fans' contribution to the Czechs' fast start should not be underestimated. How the players react to the more partisan atmosphere that will greet them when they return to this stadium for their final group game against co-hosts Poland at the weekend will be an important factor.
Cech's error may not have cost his team any points in this game but it could have had consequences. Greece pushed hard for an equaliser and that required the Czechs to battle hard to hold on to their precious lead. In tournament play, any rest that teams can get can be crucial as the competition progresses. There are some ageing legs in Bilek's side and it will be interesting to see if they fade late on against a young Polish side in Wroclaw on Saturday.
It may seem churlish to pick out the performance of a top-class goalkeeper who helped his side to Champions League glory last month - especially after his side ran out 2-1 winners here - but just what was Cech playing at for the Greece goal? The Chelsea keeper appeared to close his eyes, fearing a collision with his own defender. Human nature, perhaps, for a man who continues to wear protective headgear following a skull injury in 2006. And yet, it was still an abysmal error.