Poland benefited from a bright opening to the Group A match before a crowd of 56,826 when Robert Lewandowski headed them into the lead on 17 minutes from a lovely Jakub Blaszczykowski cross as Greece goalkeeper Kostas Chalkias paid the price for coming from his goal but failing to collect the ball as it flew across his area.
Greece had managed to upset hosts Portugal in the opening match of the tournament on their way to winning Euro 2004, but their evening was rapidly turning into a nightmare when Sokratis Papastathopoulos was red carded three minutes before half-time for a second yellow card after he was deemed to have fouled Rafel Murawski midway inside his half.
He had been booked moments earlier for a supposedly late challenge on Lewandowski despite replays showing he had won the ball fairly and squarely in the air against the Borussia Dortmund striker.
It was a harsh decision by referee Carlos Velasco Carballo, but perhaps not a major surprise when one learned later that the Spanish official had produced 16 red cards in 19 La Liga matches last season.
Poland looked to be heading for a momentous first win at the European Championships, but were stunned on 51 minutes as Arsenal goalkeeper Wojciech Szczesny missed Vassilis Torosidis's cross ball enabling Greece substitute Dimitrios Salpingidis to stab the ball home from close range.
Szczesny's night went from bad to worse when he was red carded on 68 minutes after felling Salpingidis as he raced in on goal on 68 minutes.
Greece captain Giorgos Karagounis stepped up to take the penalty, but planted his penalty too close to substitute Polish goalkeeper Przemyslaw Tyton who dived to his left to make the stop and ensure his side a point from a match they might have won given their advantage at the break.
Whether it will prove to be good enough to help Poland reach the last eight of the tournament remains to be seen with Russia and Czech Republic their other opponents in what appears to be the weakest group at the finals.
It was all Poland in the first period with Chalkias forced to swipe an effort by Murawski over his bar on four minutes with Lewandowski just failing to connect with a Lukasz Piszcezk cross eight minutes later.
Damien Perquis smashed a shot wide when he should have hit the target eight minutes before half-time as Poland could and should have finished matters off ahead of the second half.
Greece looked a whole lot more sprightly as they re-emerged with their 10 men as Salpingidis - on for Sotiris Ninis at half-time - made a noticeable difference to their attacking instincts. It was hard to argue they were not worthy of the point.