Ireland made seven changes from their opening game, with Max Flynn and Ruben Moloney making their Under-20 debuts.
They made a bright start to the game as Hugh Gavin scored an early try on 11 minutes with Sean Naughton adding the extras.
Molony then had another try chalked off after a TMO review.
Stephen Smyth did eventually get over the line for their second try on 20 minutes, with Naughton off-target from the tee.
Luka Tsirekidze reduced the deficit for Georgia with two penalties before the break.
Georgia, who had lost 35-11 to Australia in their opening match, made a storming start to the second half and took the lead for the first time with a try five minutes after the restart.
Nika Lomidze crossed with Tsirekidze converting to give his side a one-point lead.
Naughton and Tsirekidze traded penalties ahead of the dramatic conclusion to the game.
Treacy touched down in the left-hand corner after a move involving a number of phases, with replacement Jack Murphy accurate from the tee to seal the hard-fought win for Ireland.
Ireland: B O’Connor; D Colbert, S Berman, H Gavin, R Moloney; S Naughton, O Coffey; J Boyd, S Smyth, A Sparrow, J McKillop, E O’Connell (capt), S Edogbo, M Flynn, L Murphy.
Replacements: M Yarr, E Calvey, P Bell, A Spicer, B Gleeson, T Brophy, J Murphy, F Treacy.
Georgia: O Metreveli; L Khorbaladze, L Kobauri, G Khaindrava, L Keshelava; L Tsirekidze, S Jigauri; L Ungiadze, M Khakhubia, D Mtchedlidze, T Tsulukidze, D Lagvilava, L Suluashvili, A Dvali, N Lomidze (capt).
Replacements: S Kheladze, L Kotorashvili, D Kuntelia, M Tskhadadze, T Ganiashvili, M Kachlavashvili, L Takaishvili, N Kevkhishvili.
