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After a cagey opening 20 minutes, it was hosts Georgia who had the first real chance when Nikoloz Gelashvili darted in from the right and fired just wide.
Moments later, Martin O'Neill's Ireland were in front with the first goal of Euro 2016 qualifying from any side as Everton's McGeady received a pass from Toffees teammate James McCarthy before slotting home.
Georgia, who are managed by former Newcastle United man Temuri Ketsbaia, equalised in the 38th minute when Tornike Okriashvili twisted and turned 25 yards out before dipping a shot just under the crossbar past the helpless David Forde.
McGeady netted the winner on the 90-minute mark with a pinpoint left-foot curler from 18 yards after a smart turn in traffic.