Outstanding flanker Mamuka Gorgodze grabbed the only try of the game for Georgia as fly-half Merab Kvirikashvili added the conversion and five penalties, with replacement Malkhaz Urjukashvili also contributing a three-pointer.
Dan Dumbrava (2) and Florin Vaicu kicked Romania's three penalties but they will end the tournament winless after losing four from four.
Georgia, who had proved so impressive in their defeats to Scotland and England, got an ideal start as Kvirikashvili gave them an eighth-minute lead with a long-range penalty, although it was quickly cancelled out by Dumbrava.
Two soft penalties around the half-hour mark allowed Kvirikashvili to open up a 9-3 lead as firstly the chasers of a kick ahead were caught offside before Daniel Carpo needlessly played the ball on the ground.
Dumbrava replied with his second three-pointer but Kvirikashvili had the last word of the first half with his fourth penalty.
And the decisive blow was struck shortly before the hour mark as Georgia worked their way through the phases for Gorgodze to burst through a tackle and reach out to score, with Kvirikashvili adding the extras.
Kvirikashvili added another penalty with 10 minutes to go to put the result beyond doubt.
Vaicu kicked Romania's third penalty but the Georgians deservedly rounded off a fine victory by registering the last score through Urjukashvili's penalty.
Georgia, coached by Scotland's Richie Dixon, could do their boss' fellow countryman a favour if they can cause a major upset against Argentina in their final pool clash at the same venue on Sunday.
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.