The match was Romania's last game and marked the end of their World Cup, while Georgia plays Argentina at the same venue on Sunday.
The first half was a forward-dominated arm wrestle as both sides tried to gain the upper hand.
Georgia showed a bit more intent to move the ball during the opening spell, but they failed to click into gear.
The boot of Georgian first-five Merab Kvirikashvili edged his team in front as the left-footed 27-year-old took advantage of Romania's continual infringing.
Things didn't open up during the first part of the second spell either as both sides showed their desperation to register a rare World Cup victory.
Georgian openside flanker Mamuka Gorgodze crashed over the line from close range in the 56th minute and Kvirikashvili's conversion gave them an winning advantage.
The pivot added another penalty and finished his night with five goals from seven attempts, while his Romanian opposite Marin Danut Dumbrava scored all of his team's points with three penalties.
Veteran Romanian hooker Marius Tincu earned his 50th cap during the game.
Georgia 25 (Mamuka Gorgodze try; Merab Kvirikashvili 5 pens, con, Malkhaz Urjukashvili pen)
Romania 9 (Marin Danut Dumbrava 3 pens)
HT: 12-6
GEORGIA 25
ROMANIA 9