Midfielder David Targamadze put the hosts ahead and the score remained 1-0 until 11 minutes from time when substitute Giorgos Fotakis equalised.
Angelos Charisteas then scored the winner for the visitors five minutes from time.
The result allowed Greece to finish top of Group F with 24 points, two points clear of Croatia, who beat 2-0 Latvia in Zagreb.
Greece had put their destiny back in their own hands with their 2-0 win against the Croats on Friday, but they were punished for a cautious start here when Targamadze beat Alexandros Tsorvas in the visitors' goal with a long-range free-kick.
Greece then poured forward in search of an equaliser only to find Georgia 'keeper Nukri Revishvili in fine form.
Their evening was in danger of turning into a massive anti-climax until goalscorer Targamadze was forced off injured 20 minutes from time.
Home coach Temuri Ketsbaia had already made all three substitutions available to him so Georgia were forced to play the remainder of the game with ten men.
The Greeks went on to take full advantage, with Fotakis' precise shot from just outside the box levelling the scores.
The stage was then set for Charisteas—who scored the winning goal against Portugal in the 2004 European Championship final—to win the game and seal his country's place in the finals with a close-range finish at the end of a well-worked move.