Substitute keeper Przemyslaw Tyton was the hero for Poland in Warsaw, where both sides finished with 10 men.
Tyton came off the bench in the second half after Szczesny had been sent off for bringing down Greek substitute Dimitris Salpigidis.
The reserve goalkeeper promptly saved the resulting spot-kick from Greece captain Giorgos Karagounis to ensure the points were shared in a wonderfully entertaining match at the National Stadium.
Poland will be kicking themselves, however, after throwing it away having gone in at half-time with a one-goal lead and with Greece down to 10 men after defender Sokratis Papastathopoulos was sent off harshly for two innocuous yellow cards.
The Poles, who began the game with swaggering intent in front their home fans, had dominated the first half.
It was no surprise when Robert Lewandowski gave them the perfect start after 17 minutes, although the goal had as much to do with the poor positioning of Greek goalkeeper Kostas Chalkias as the sharpness of Lewandowski.
Chalkias found himself flapping his arms in no man's land to leave the Borussia Dortmund striker, who scored a hat-trick in the German cup final last month, to head home a swinging cross from Jakub Blaszczykowski.
It was no more than the Poles deserved.
They were hungrier, quicker to the ball and seemingly desperate to prove that while they are the lowest-ranked team in the tournament they have a realistic chance of reaching the quarter finals at least.
In that 45 minutes the Greeks looked a shadow of the organised side who had gained a reputation for being tough to beat, having lost just one of 21 games under manager Fernando Santos.
In fact, the Poles would have been kicking themselves that they were not further ahead at half-time after Rafal Murawski, Maciej Rybus and Lewandowski all missed clear-cut chances.
Damien Perquis should also have scored when the ball fell to him in the penalty area but the Polish defender, in unfamiliar territory, drilled his shot wildly wide of the post.
It looked as if the Greeks' fortune had deserted them completely, however, when Papastathopoulos saw red just before half-time.
The Greek defender had become the first player to be booked in the tournament when he received the yellow card from Spanish referee Carlos Velasco Carballo for what appeared an harmless challenge with a Polish attacker.
His second yellow was equally as harsh when he was adjudged to have brought down Murawski. The referee, however, produced red and Greece were down to 10 men and seemingly heading to an opening defeat.
Perhaps fired up by the injustice the Greeks, however, came out in the second half with more determination and resolve and it paid off, even in they did have Szczesny to thank for their equaliser.
Szczesny raced out of his goal attempting to collect Vassilis Torosidis' right-wing cross but misjudged it totally, allowing substitute Salpigidis to score from close-range.
The drama, however, was far from finished and the teams were down to 10 players each after 70 minutes when Szczesny endured a moment of madness, sticking out a leg to trip Salpigidis in the penalty area.
The resulting penalty was delayed while PSV Eindhoven goalkeeper Tyton came off the bench and Maciej Rybus went off, but Tyton's first action was the most crucial of the match, saving the tame spot-kick of Karagounis.
A point was probably a fair result. But what a dramatic way for Euro 2012 to begin.