Levan Kobiashvili struck in the last minute to condemn Croatia to their first Euro 2012 qualifying defeat and maintain Georgia's superb form under coach Temuri Ketsbaia.
The hosts extended their unbeaten record to 10 games under Ketsbaia as Kobiashvili lifted them onto nine points in Group F, just a point behind Croatia who led the way heading into today's round of fixtures.
The result opened the door for Greece to move to the top of the pile providing they could secure victory in the late kick-off away to Malta.
Croatia bossed the opening half and spurned a chance to go in front on 26 minutes when Darijo Srna's inswinging corner was headed wastefully over by Dejan Lovren.
Two minutes later Ivan Rakitic's shot from the edge of the box was saved by Nukri Revishvili, while the Georgia goalkeeper also had to be on his toes to deny Nikola Kalinic's long-range effort seven minutes before the interval.
Niko Kranjcar's 20-yard effort again tested Revishvili three minutes after the restart, while Hertha Berlin midfielder Kobiashvili had Georgia's first chance of the night moments later when he fired over.
Croatia continued to make most of the running with Ivan Perisic blazing the ball over the bar from outside the area on 72 minutes, but the closing stages went the way of Georgia as the visitors ran out of ideas.
Otar Martsvaladze went close for the hosts with an effort that Vedran Runje saved with his feet, but with the clock ticking down and a draw the most likely outcome, Georgia mounted one last attack to secure all the points.
Martsvaladze finally unlocked the Croatia backline, his pass from the left wing picked up by the advancing Kobiashvili who lashed the ball left-footed into the roof of the net from just inside the area.