Georgia defender Akaki Khubutia's first-half own goal handed Scotland an unconvincing victory in their European Championship qualifier.
Inspired by the tenacity of stand-in captain Scott Brown, the Dark Blues clung on to claim their first victory in Group D.
Scotland rode their luck against Temuri Ketsbaia's resolute sidewhen the Georgia goalkeeper, Giorgi Loria, pushed Shaun Maloney's 28th-minute strike on to team-mate Khubutia's knee and watched helplessly as the ball bounced past him into the net.
The Scots should have cemented their lead in the second half but had to see out a nervy last 10 minutes after Everton striker Steven Naismith missed two clear chances.
The Scotland manager, Gordon Strachan, however, called the performance a "manager's dream".
He said: "In the first half you're thinking, 'must win and a good performance would be good as well', but to get a great performance after losing the first game with the pressure on, to play that way was absolutely outstanding.
"We practised most days of the week to play against three men at the back and then they switched to four so to reorganise and play such stunning football is really impressive.
"To create that many chances against a good side, who can break away to cause you problems, and not just throw the ball in but have lots of variation, that's a manager's dream."
Strachan dropped his captain Darren Fletcher, who has seen just 25 minutes of Barclays Premier League action for Manchester United since leading the national team out in Germany last month. Strachan decided he had better options in Brown, who returned after missing the game in Dortmund with a hamstring tear, and Shaun Maloney, who came in along with the Hull full-back Andrew Robertson and Sunderland's Steven Fletcher.
Those who stumped up £45 for a ticket made an almighty din as the game got under way.
Maloney created the first opening 18 minutes in as he clipped a short cross towards Naismith, whose deft flick failed to find the target.
Scotland went close again as a slick move ended with Maloney teeing up Anya in the box, only for Khubutia to block the winger's strike at close range. But, after Grant Hanley just failed to get on the end of Naismith's flick, Strachan's team grabbed the lead from the resulting corner, .
Robertson's kick was punched clear by Loria, who got down well to save Maloney's follow-up shot.But the loose ball bounced off Khubutia and into the net.
Brown thought he had added a second seven minutes before the break whenhis shot from 15 yards deflected wide, while Fletcher could headed over a Robertson cross.
Hanley's hesitancy handed the visitors hope but Nikoloz Gelashvili – yet to score in his 20 caps – blazed over from a poor angle.
Yet there remained a nervousness about the Scots, and Naismith fired straight at Loria after Fletcher put his strike partner through.
With 13 minutes left, Naismith wasted another must-score chance. Fletcher knocked a clever back heel into the path of Naismith, who pushed his shot past the far post.
So the Everton striker was relieved to see Irakli Dzaria drag a 15-yard shot wide seconds later.
The Scots could afford to see Loria save again from Morrison as they claimed a much-needed win.
Scotland (4-3-3): Marshall; Hutton, Martin, Hanley, Robertson; Maloney, Brown, Anya; Fletcher (Martin, 89), Morrison, Naismith (McArthur, 80).
Georgia (4-4-2): Loria; Lobjanidze, Khubutia, Kvirkvelia (Okriashvili, 46), Kverkvelia; Daushvili, Grigalava, Kazaishvili (Chanturia, 80), Kankava; Gelashvili, Papava (Dzaria, 70).
Referee Miroslav Zelinka (Cz Rep).