The loss consigns the Romanians to bottom place in their pool and marks only the second time in their seven World Cup appearances that they have failed to win a match at the tournament.
Despite both sides claiming it was their own final, the match never reached any great heights with both teams intent on spoiling their opponent's ball to halt any momentum and accepting the consequences.
Referee Dave Pearson was an influential figure, awarding 14 penalties in the first half.
Georgia flyhalf Merab Kvirikashvili took full advantage of Romania's transgressions in the first half, slotting four penalties to give his side a 12-6 lead at the break after Romanian flyhalf Marin Danut Dumbrava was successful with two of his three penalty attempts.
Both sides missed early second half penalty attempts before the Georgians finally built some phases through quick ball that led to Gorgodze crashing over for a 56th minute try, which Kvirikashvili converted to give his side a 19-6 lead.
The flyhalf put the result beyond doubt when he added his fifth penalty before Romania's Florin Adrian Vlaicu slotted his side's third penalty of the match.
Georgia replacement Malkhaz Urjukashvili completed the scoring with another penalty in the dying minutes.