Lewandowski's header just before halftime in the Group D match was his 15th goal in the last six games for his club Bayern Munich and country.
He also equalled Northern Ireland forward David Healy's scoring record for a Euro qualifying campaign with 13 goals.
Ireland, who finished with 10 men after captain John O'Shea was shown a second yellow card, needed to win or draw by at least 2-2 to steal second spot behind world champions Germany.
Poland went ahead in the 13th minute as midfielder Grzegorz Krychowiak fired home from outside the area after a corner.
Jon Walters equalised for Ireland three minutes later after defender Michal Pazdan brought down Shane Long.
"I have to tell you I'm so proud," Lewandowski shouted through a microphone to fans wearing a red and white "Vive la Pologne 2016" T-shirt while his team mates uncorked champagne on the pitch.
"The Irish team did not create any real opportunities. We deserved to advance, we were the better team and played a good qualifying round," he told Polsat TV.
Poland co-hosted the 2012 tournament with Ukraine and will be playing in their third straight Euro finals.
Ireland, who finished third on 18 points behind Poland with 21 and Germany on 22, head for the playoffs hoping to replicate the form they showed in beating Germany 1-0 earlier this week.
"I think the message is still pretty clear. We're still there but we have two games to go," coach Martin O'Neill said.
"The win with Germany gave us a lift, but we didn't have the same intensity. Players were ready, but it was hard to replicate it away against a good Polish side."
Asked if Lewandowski could win the Ballon D'or world footballer of the year award, he said: "Yeah, very possibly. A very fine player. I say that he is a world class player and he knows the game. Poland must be delighted to have him."
(Editing by Patrick Graham and Ken Ferris)
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