The Everton midfielder left it late to snatch victory for his side in Tbilisi, turning on the edge of the area before firing home his second goal of the game in the last minute to get the Irish up and running in Group D.
He had put the visitors ahead in Sunday's encounter with a well-taken 24th-minute goal before a wonder strike from Tornike Okriashvili looked to have guaranteed Georgia a share of the spoils when he let fly from distance shortly before half-time.
Ireland were far from at their best for long spells and they let their tempo drop in both halves.
But McGeady's late intervention gave boss Martin O'Neill victory in his first competitive game in charge, and crucially a platform on which to build their qualifying campaign.
Ireland reverted to a more familiar line-up after naming an experimental XI in Wednesday's 2-0 friendly win over Oman in Dublin.
And it was O'Neill's side who controlled possession in a lacklustre opening 20 minutes in which Georgia appeared happy to defend deep and attempt to counter-attack.
Georgia's tactic almost proved successful when Nikoloz Gelashvili capitalised on sloppy defending by Marc Wilson to race through on goal, but his shot from the right was deflected wide of the far post.
From the resulting corner, Akaki Khubutia's glancing header narrowly evaded the stretching Ucha Lobzhanidze.
Ireland almost immediately punished those missed opportunities with a swift counter-attack of their own to break the deadlock in the 24th minute.
James McCarthy did well to control Jonathan Walters' knock down and the Everton midfielder then slid a pass into the path of club-mate McGeady, who kept his nerve to slot past the onrushing Giorgi Loria.
The visitors appeared comfortable after their opener, but Georgia levelled seven minutes before the break with a breathtaking finish from Okriashvili.
Ireland were caught cold as the midfielder turned Stephen Ward before lashing his shot past a helpless David Forde into the top right-hand corner.
Loria was injured at the end of the first half in a collision with Stephen Quinn and failed to return after the break, with Roin Kvaskhvadze taking his place between the posts.
Ireland continued to enjoy the lion's share of possession in the second half but struggled to find a cutting edge and were often wasteful with the ball.
McGeady proved to be their most potent weapon, the winger shooting just wide from a well-worked set-piece before he then curled a shot straight at Kvaskhvadze from the left.
O'Neill introduced Shane Long and Robbie Brady for the final 15 minutes in an attempt to boost Ireland's attacking threat, but it was McGeady who was the difference.
In the final minute, the 28-year-old brought the ball down and turned sharply on the edge of the box, before showing quick feet and bending an accurate strike into the top corner.
O'Neill can now focus on next month's clash against Gibraltar on the back of a win, while Georgia must regroup ahead of their trip to Scotland.
