The Everton winger had given the visitors the lead early in the first half after great work from skipper Robbie Keane in a fine move at the Boris Paichadze Dinamo Arena on Sunday.
However, a real bolt out of the blue from the lively Tornike Okriashvili levelled proceedings just before the break which seemed to take the wind out of the visitors’ sails.
Yet with the game seemingly petering out, McGeady produced a moment of briliiance to give Martin O’Neill three points in his first competitive match in charge, and momentarily send the Green Army top of Group D.
O’Neill surprised a few by naming Hull midfielder Stephen Quinn in the centre of his midfield five, with Stoke striker Jon Walters lining up on the right and McGeady occupying the left-wing berth earmarked for James McClean before he was ruled out by injury.
Central defender Solomon Kverkvelia had to hack away a first-minute Walters cross at full-stretch, and Guram Kashia had to be equally alert 18 minutes later after McGeady had tricked his way past wing-back Ucha Lobzhanidze and sent a teasing ball into the middle.
Georgia gradually eased their way into the game and after David Forde, preferred in goal to 38-year-old Shay Given, had managed to get just enough on Nikoloz Gelashvili’s rasping drive, Lobzhanidze came close to connecting with Kverkvelia’s flick-on at the far post.
But they fell behind within seconds when, after Walters climbed into an aerial challenge, Robbie Keane allowed the ball to run to James McCarthy and he drew his man before sliding McGeady in to finish with ease.
O’Neill and assistant Roy Keane leapt from the bench in celebration – but their joy was to be short-lived.
Skipper Jaba Kankava saw a 34th-minute strike deflected just wide but Georgia were level within four minutes when Okriashvili picked up possession on the right and after turning away from John O’Shea – the only survivor from Ireland’s last visit to the city in 2003 – and then Stephen Ward, thumped a dipping effort over Forde and into the net.
Georgia returned determined to regain the momentum with which they had ended the first half and committed men to the search for an early lead.
Robbie Keane hacked away an Okriashvili effort in the 52nd minute, then Forde too had to be on his toes to clutch Jano Ananidze’s low cross to his chest seconds later with Gelashvili looming.
But it was opposite number Roin Kvashvadze, a half-time replacement for Giorgi Loria, who threw himself desperately to his left as McGeady curled a shot inches wide from a well-worked 57th-minute free-kick.
Keane scuffed a volley harmlessly out of play at the far post, before McGeady cut inside and forced Kvashvadze to pluck the ball from underneath his crossbar.
However, as the game entered the closing stages, it was the Georgians who were building up a head of steam, and O’Neill responded with 14 minutes remaining by replacing Quinn and Keane with Robbie Brady and Shane Long in a bid to add new vigour to his attack.
In the meantime, McGeady had squandered yet another opportunity when he lashed at a dropping ball on the edge of the penalty area, slicing it wide when greater composure might have served him better.
However, he made no mistake when presented with another chance with seconds of normal time remaining, accepting Seamus Coleman’s pass before stepping on to his left foot and curled home a precious winner.
The Republic host Gibraltar in Dublin in their next Group D game on October 11, before heading to Gelsenkirchen to take on Germany three days later.