The Poles needed a second-half penalty save by little-known goalkeeper Przemyslaw Tyton to earn a point despite a rousing start.
"We were really prepared and you could see the effects of all our training sessions," Smuda told a news conference. "Perhaps the team had too much pressure. The team is young."
Smuda said Poland, playing in front of a capacity home crowd of 56,000 fans at the National stadium, became too defensive after Greece lost Sokratis Papastathopoulos to a red card shortly before halftime.
"We wanted to maintain the [1-0] score. We played sideways and backwards," he said. "The Greeks had nothing to lose."
Smuda praised replacement keeper Tyton, who saved Giorgos Karagounis's penalty with his first touch 20 minutes from time after Wojciech Szczesny was sent off for bringing down Dimitris Salpingidis.
"Tyton, at the clubs where he played was really spectacular at [saving] penalty kicks," Smuda said.
Striker Robert Lewandowski, who scored a 17th minute opener and was a constant menace to the Greek defence, said the Poles showed they were good enough to reach the quarter-finals.
"We're still in the game. I hope that if we play like we played in the first half, we could win the next two games," he said.
Poland, the lowest ranked team at the tournament, face Russia in the capital on Tuesday.
"I congratulated the players on this game," Smuda said. "I told them cheer up because it would be much worse if we had lost. We should not dwell on this result."
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