Coleman, 27, made a strong run down the right, cut into the box and slotted home after the ball ricocheted into his path thanks to two fortunate rebounds.
It was a victory which the Irish barely deserved from a match which produced little creativity and goalmouth action.
The Republic are one of four Group D teams with four points from two games.
Austria and Wales drew 2-2 in Vienna and Serbia won 3-0 in Moldova in Thursday's other two Group D encounters - suggesting this could be a very competitive group.
For Martin O'Neill's men, this match and Sunday's game with Moldova represent an important few days in their quest to reach the finals in Russia, with anything less than six points a blow to their qualification hopes.
With only the group winners guaranteed an automatic place in the finals, the chance to pick up maximum points against the supposed two weakest teams in the group is crucial to Ireland's ambitions of qualifying for a first World Cup finals since 2002.
The Republic had won all of their previous seven contests with the Georgians, six of them in competitive fixtures, including two one-goal victories in the qualifiers for Euro 2016.
But on Thursday, Georgia, who had lost 2-1 at home to Austria in their opening group game, posed the greater threat in a largely uninspiring first half.
Valeri Kazaishvilli turned and fired a left-foot shot wide of the post in the 17th minute and then the visitors struck the woodwork twice in quick succession with eight minutes of the half remaining.
Levan Mchedlidze thundered a header off the crossbar and when the home side failed to clear, the ball fell to Guram Kashia, whose looping header back over Darren Randolph bounced off the post.
Under-strength Republic battle to three points
The game was the Republic's first since O'Neill signed a deal to continue as manager until the end of the present campaign - and their first competitive home game since reaching the last 16 of Euro 2016.
They started their qualifying campaign with a 2-2 draw in Belgrade last month but Daryl Murphy, the man whose late goal earned them a draw against Serbia, was among eight players ruled out through injury.
Goalkeeper Keiren Westwood, Harry Arter, Stephen Quinn, Kevin Doyle, Aiden McGeady, Anthony Pilkington and Marc Wilson were also absent.
And against a stubborn Georgian side it took an improved second-half performance for O'Neill's team to secure a third win in their last 10 matches.
The Irish showed greater intensity after the resumption and Coleman's strike gave them the boost they needed but scarcely deserved on the balance of play.
James McClean had a header ruled out for offside and the lively West Brom winger later saw another effort cannon off the underside of the bar during a lengthy spell of injury-time caused by a head injury to Robbie Brady, who was forced to retire hurt after receiving treatment following a clash of heads with Georgia defender Solomon Kverkvelia.
The Football Association of Ireland later said Brady was conscious in the dressing room but went to hospital as a precaution for a scan.
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