In Nice, Arkadiusz Milik's goal in the 51st minute helped Poland to edge newcomer Northern Ireland 1-0 in Group C.
It was the first-ever win for Poland at a European Championship, following two disappointing displays in the last two tournaments.
"It's just the beginning, the next matches will show that we can play even better," said Poland midfielder Grzegorz Krychowiak.
Lewandowski, Poland's top star, scored 13 in the Euro 2016 qualifiers, was besieged by defenders of Northern Ireland when he touched the ball and had no chance to score.
In Paris, a thumping volley from Modric helped Croatia earn a 1-0 victory over Turkey in its Group D opener.
The 30-year-old sent in the dipping volley with his right foot in the 41st minute.
"It was one of his (Luka's) best matches in Croatia's history. He was really playing well. He was really our leader," said Croatia coach Ante Cacic.
"He scored a magical goal and at the end it was the only goal. We need Luka in this form. All the team looks better and more aggressive when he plays like that," Cacic added.