The Pumas had been trailing 12-9 at half-time but Moyano ensured the home side surged ahead in the second half, with Tomas Cubelli also scoring a late unconverted try.
Beka Tsiklauri scored all but three of Georgia's points from the penalty tee, with Lasha Malaguradze taking over kicking duties late in the game.
However, despite two penalty misses by Bustos Moyano and Benjamin Madero failing to add the extras to Cubelli's 77th-minute try, Argentina claimed the win against the second tier side.
It looked like it would be Georgia's best chance to secure a win against a tier one country when they went into the interval ahead.
Montpellier utility back Bustos Moyano opened the scoring in the third minute before two penalties from Tsiklauri sandwiched a miss from the Argentinian as Georgia moved ahead.
Bustos Moyano levelled the score again but he then missed again while his Georgia goalkicking counterpart scored twice to edge ahead at the break.
The second half started with Bustos Moyano kicking three penalties while Tsiklauri kept Georgia in contention with a drop goal.
Argentina then pulled away with two more successful shots at goal before Malaguradze took over from Tsiklauri and added a further three points for the Georgians.
Cubelli then took the game out of Georgia's grasp with a late score even though Madero was unable to make the conversion.