Georgia raced into a 13-0 lead but after a nervous opening 30 minutes, Portugal took control of the game in the second half and scored 18 unanswered points heading into the final two minutes of the contest.
But when replacement scrumhalf Pedro Lucas gave away a needless penalty for offside at the ruck, Georgia were able to set up an attacking lineout and drive over through hooker Tengiz Zamtaradze.
Luka Matkava missed what would have been the winning conversion from out wide, and later Guedes put his penalty attempt off target with the final play of the game in a frenetic and hugely entertaining finish to the match.
"The first half was so poor," Portugal coach Patrice Lagisquet said. "We were making mistakes in the lineout and we weren't disciplined.
"At halftime, we were lucky they led by eight points and we are still in it. We can't have such a bad half, let's play your rugby. Be yourself. With that first half, the result is a good one for us."
Georgia have now gone 19 games unbeaten against Portugal, though there are four draws in there, as winger Akaki Tabutsadze and Zamtaradze crossed for their two scores.
The excellent Raffaele Storti grabbed a brace of tries for Portugal, who will rue their inability to manage the final few minutes, and Lucas’ rush of blood to the head that handed Georgia a scoring chance.
"That’s rugby, it isn’t one half," Georgia coach Levan Maisashvili said. "We played really hard in the first half but in the second Portugal dominated everywhere.
"We know their team but we didn’t execute our opportunities when we had a chance. We lost a lot of balls today and lost at the breakdown."
Portugal had a chaotic start as they conceded the opening try inside two minutes when Georgia spun the ball out wide and Davit Niniashvili gave a lovely slight-of-hand pass to Tabutsadze, who dotted down.
That rattled the Portuguese, who were rushed into errors in their own 22 and only let off the hook by a Georgian side that were unable to punish them.
Flyhalf Tedo Abzhandadze landed the conversion and two penalties to stretch the Georgian advantage, before Portugal showed a glimpse of their potential shortly after the half-hour mark.
They have an exciting back three full of running and Storti spotted the smallest of gaps in the Georgian defence and raced clear to score in the corner.
That momentum was halted momentarily by a desperately poor piece of play from prop Francisco Fernandes, who made no attempt to wrap his arms in a tackle on opposition hooker Shalva Mamukashvili and was deservedly sent to the sin-bin by referee Paul Williams as Georgia led 13-5 at halftime.
Portugal rode out the 10 minutes a man light and scored the first points of the second half through two Samuel Marques penalties.
They were well on top and with Georgia unable to get out of their half, Storti scored his second, fed by a brilliant offload in the tackle from Jeronimo Portela to make it 18-13.
Georgia finally got a foothold in the Portugal half in the closing stages and when Lucas gave away his needless penalty, they were able to set up an attacking lineout and drive over through Zamtaradze to level the score.
