The visitors looked sharp during the early stages, but it was Georgia who had the first chance when Valeri Kazaishvili lashed an effort wide.
Scotland continued to dominate possession for the remainder of the half, though they struggled to create clear-cut opportunities in the final third.
Robertson managed to whip an incisive ball into the box on 29 minutes that Fletcher was inches away from getting on the end of as Scotland threatened to take the lead.
The game's opening goal arrived on 38 minutes as Georgia struck to go ahead. A chipped ball into the box was headed down by Levan Mchedlidze before Kazaishvili pounced to hit a low drive past David Marshall.
Georgia came close to doubling their advantage at the start of the second half when Jano Ananidze's header from a Kazaishvili cross was well held by Marshall.
Scotland enjoyed plenty of the ball after that, but an inability to convert their dominance into goalscoring opportunities gave the home side the chance to pose a threat on the break.
Marshall was called into action once again with 68 minutes on the clock as the Cardiff City shot-stopper was forced to tip Mchedlidze's stinging effort away from danger.
Mchedlidze was in the thick of it again soon after, though this time he smashed the ball narrowly off target as the Scots just about held on.
Scotland piled on the pressure during the closing stages as they frantically searched for an equaliser, but Georgia's resolute defence did enough to keep them out.