Replacement Luka Matkava kicked a 78th-minute penalty to prove the Georgian hero.
Wayne Pivac's side had built up a 12-3 lead thanks to two tries from Jac Morgan.
But Georgia responded in the second half with an Alexander Todua try and the boot of Matkava to create history and spark wild celebrations.
The defeat, in a year in which Wales have also lost at home to Italy, will bring Pivac's position under intense scrutiny, less than a year before the World Cup in France.
Georgia had defeated Italy in the summer, but this result will further strengthen the calls for them to be introduced into the Six Nations.
For Pivac's side, it is hard to see where they go from here.
Wales finish the Autumn Nations campaign against Australia next Saturday following a performance that fell well short of what should be expected from this international side.
A 55-23 hammering handed out by the All Blacks in the opening match was followed by a gritty 20-13 victory over Argentina that was built on determined defence.
That brought hope of a more coherent Pivac outfit, but this inept display dispelled that theory.
A wasteful Wales attacking unit was again under the microscope as they could not consistently break down the Georgian defence.
The hosts failed to score after 24 minutes and were deservedly defeated by a better team who dominated opponents that appeared to lack desire and structure.
Wales have suffered some humbling historical home defeats - including losses to Romania in 1988, Western Samoa in 1991, Canada in 1993 and Italy earlier this year. This result rivals those past defeats.
It was a first win for Georgia in four attempts against Wales and a victory 11 months before the World Cup encounter between the two sides in Nantes.
Georgia's past two visits to Wales saw them proved dogged opponents as the home side prevailed 13-6 in 2017 and 18-0 two years ago.
Pivac made fewer changes than normally associated with this fixture, with six alterations to the side that defeated Argentina, two of those injury enforced following the loss of Will Rowlands and Dan Lydiate.
Wing Josh Adams returned from injury with fly-half Rhys Priestland and centre Owen Watkin starting while Ben Carter, Morgan and the uncapped Josh Macleod were included in the pack.
Macleod, normally a flanker, made his Test debut at number eight to finally gain his first cap after a long wait.
The 26-year-old had been picked to start against Scotland in the 2021 Six Nations, only to rupture his Achilles tendon in training a few hours later.
Wales, who were wearing black armbands to mark the death of former coach John Ryan, made a poor start from the Georgia kick-off and conceded an early penalty with Tedo Abzhandadze slotting over the kick.
The hosts replied from a well-worked training-ground move involving locks Adam Beard and Carter before Morgan stepped his way over unopposed for a first international try. Priestland converted from the touchline.
Morgan again popped up on the right-hand touchline for a second score after benefiting from incisive build-up work from North, Cuthbert and Tomos Williams.
Cardiff scrum-half Williams was almost involved in a brilliant third try for Wales with his basketball-style overhead pass finding Adams who chipped ahead and sprinted away. His effort was in vain, though, as Williams' pass was adjudged to have been forward.
Wales led 12-3 at half-time and would have been frustrated they were not further ahead - opportunities were spurned with wasteful kicks, and there was a reliance on a formulaic attacking approach.
Wasteful Wales continued to be unconvincing in the attacking third and wing Cuthbert was shown a yellow card for an aerial challenge on Todua while chasing a high ball.
The Cardiff crowd of 63,585 gave their verdict on the entertainment value on show as they produced their own Mexican wave.
Macleod was replaced by Taulupe Faletau, who was playing his 99th international for Wales and the British and Irish Lions.
Faletau was soon followed onto the field by Exeter lock Dafydd Jenkins as the 19-year-old lock won his first cap.
Georgia took advantage of their numerical advantage of Cuthbert's absence from a well-worked try for wing Todua who took a clever kick from Abzhandadze, who also converted to reduce the deficit to two points.
When Cuthbert returned, Wales also took off Adams and brought on full-back Leigh Halfpenny, back for his first international for 16 months because of a serious knee injury.
Georgia were rampant and dominating the aerial battle with Abzhandadze missing a penalty chance to give Georgia the lead.
Morgan momentarily thought he had scored a hat-trick, but Faletau knocked the ball on in the build-up.
The Georgia scrum started to dominate and forced the final penalty which Matkava slotted over.
It left the visitors to celebrate what they undeniably deserved and Pivac to ponder what his future might be.
But to Georgia go the plaudits. And rightfully so.
Wales: Rees-Zammit; Cuthbert, North, Watkin, Adams; Priestland, T Williams; G Thomas, Owens, D Lewis, Carter, Beard, J Morgan, Tipuric (capt), J Macleod.
Replacements: B Roberts, R Jones, S Wainwright, D Jenkins, Faletau, Blacker, Costelow, Halfpenny.
Georgia: Niniashvili; Modebadze, Tapladze, Sharikadze (capt), Todua; Abzhandadze, Lobzhanidze; Gogichashvili, Chkoidze, Papidze, Cheishvili, Mikautadze, Gorgadze, Saghinadze, Jalagonia.
Replacements: Mamukashvili, Abuladze, Kuntelia, Chachanidze, Mamamtavrishvili, Matkava, Khmaladze, Lomidze.
Referee: Andrea Piardi (FIR)
Assistant Referees: Pierre Brousset (FFR), Gianluca Gnecchi (FIR)
TMO: Brendon Pickerill (NZR).