Borussia Dortmund striker Lewandowski put the co-hosts in front but substitute Dimitris Salpingidis equalised for the Greeks.
Both sides ended the game with ten men, as Sokratis Papastathopoulos and Wojciech Szczesny were dismissed.
The Poles made the stronger start to the game and they had the first sight of goal in the sixth minute.
After a surging run from full-back Lukasz Piszczek, the ball fell to Rafal Murawski, and his 20-yard effort was turned over the bar by goalkeeper Kosta Chalkias.
The co-hosts continued to threaten and they nearly took the lead with 14 minutes played. Piszczek’s dangerous cross caused confusion and was just inches away from finding club team-mate Lewandowski’s forehead.
It didn’t take long for the Poles to turn their superiority into a goal as they netted the opening goal of the tournament moments later.
Polish skipper Jakub Blaszczykowski delivered a teasing centre, which Greek keeper Chalkias got nowhere near, and Lewandowski, who netted 30 goals for the German champions last season, planted a header into the empty net.
Greece looked a shadow of the side that shocked Europe by winning the trophy in 2004, and Giorgos Karagounis’ wild strike on 24 minutes epitomised their poor start to this match.
With eight minutes left before the break, the White Eagles wasted a great opportunity to double their advantage. After a set-piece was only half-cleared, defender Damien Perquis screwed his shot wide of the post.
Things got worse for Fernando Santos’ team as the half-time interval approached, with Papastathopoulos harshly sent off by Spanish referee Carlos Velasco Carballo.
After failing to cause any danger to the home defence in the first half, Santos’ side pulled themselves level on 51 minutes.
Full-back Vassilis Torossidis swung a cross in from the right-hand side and after Arsenal keeper Wojciech Szczesny didn’t claim, the ball deflected to the feet of substitute Dimitris Salpingidis, who easily poked home.
The Greeks had lost all of their eight previous encounters on Polish soil but they were handed a great opportunity to change that on 68 minutes.
Kostas Fortounis’ through-ball found Salpingidis in acres of space, and after he was brought down by Szczesny, the referee pointed to the spot.
Replacement Przemysław Tytoń made himself an instant hero with the Polish faithful though, as he dived to his left to keep out Karagounis’s tame penalty.
Despite both sides valiantly searching for a winning goal, it didn’t come. Incredibly, this meant that for the third straight European Championships, the host nation failed to notch a victory in their opening match of the tournament.
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