Shortly after substitute Dimitris Salpigidis had equalized for Greece by knocking in a loose ball in the 51st minute, the diminutive striker was taken down by Poland goalkeeper Wojciech Szczesny in the area.
Szczesny was given a red card, and captain Giorgos Karagounis was given the chance to again give Greece an opening match win against the host. Eight years ago, Karagounis scored in the opening 2-1 win over Euro 2004 host Portugal.
His shot, however, was saved by replacement goalkeeper Przemyslaw Tyton, whose first touch of the ball salvaged a point for the co-hosts in Group A.
Salpigidis came on at halftime, a gamble for Greece coach Fernando Santos to leave three attackers on the field after the loss of both central defenders in the first half - Sokratis Papastathopoulos to a red card after a pair of yellows, and Avraam Papadopoulos to injury.
But Salpigidis didn't disappoint.
Greece midfielder Vassilis Torosidis, who dropped back into central defense after the ejection of Papastathopoulos, crossed the ball into the center for Fanis Gekas.
Gekas and Poland defender Marcin Wasilewski collided and Szczesny jumped for the ball, but failed to clear it. That left the loose ball sitting patiently as Salpigidis ran in to knock it into the net.
Poland controlled the play in the first half, exploiting the right side on the field and eventually taking the lead in the 17th minute.
Two of the Borussia Dortmund trio that work on the right side set up the goal. Midfielder Jakub Blaszczykowski crossed the ball into the center for Robert Lewandowski, and the striker calmly headed the ball into the ground and under the charging Costas Chalkias.
After beating Portugal in the opening match of Euro 2004, the Greeks then beat the Portuguese again in the final to win their only major title.
Poland: Wojciech Szczesny, Sebastian Boenisch, Marcin Wasilewski, Damien Perquis, Lukasz Piszczek, Maciej Rybus (Przemyslaw Tyton, 70), Eugen Polanski, Ludovic Obraniak, Rafal Murawski, Jakub Blaszczykowski, Robert Lewandowski.
Greece: Costas Chalkias, Vassilis Torosidis, Sokratis Papastathopoulos, Avraam Papadopoulos (Kyriakos Papadopoulos, 37), Jose Holebas, Yiannis Maniatis, Giorgos Karagounis, Costas Katsouranis, Sotiris Ninis (Dimitris Salpigidis, 46), Georgios Samaras, Fanis Gekas (Costas Fortounis, 68).