Anghel Iordanescu's team, who play in the tournament's opening game against hosts France next Friday, took the lead after 68 seconds when Adrian Popa punished a defensive blunder by dribbling past Georgia keeper Roin Kvashvadze to score.
The home side doubled their advantage two minutes later when Georgia captain Aleksandr Amisulashvili slid the ball into his own net from Romania captain Razvan Rat's left-wing cross.
The last time the Romanians scored twice in the first three minutes was in a 4-2 win over China in a friendly in 1984.
Romania, who have been struggling with injuries and players not featuring for their clubs, had conceded six goals in three friendlies in the last 10 days against two in Euro qualifying, and Iordanescu was determined to tighten the defence.
On the back of a 4-3 loss to Ukraine he fielded Cosmin Moti in the centre of defence instead of Vlad Chiriches after the Napoli defender's disappointing performance on Sunday and they were never under any serious or sustained pressure.
Georgia, ranked 137th in the world, barely registered a shot on goal and allowed the home side to dictate the play, with the hosts hitting the post twice midway through the first half.
Florin Andone got a chance up front, having missed Romania's training camp in Italy and their last two friendlies against DR Congo and Ukraine due to club commitments with Spanish second-tier side Cordoba for whom he scored 21 goals last season.
But it was Nicolae Stanciu who made it 3-0 after the break with his fourth goal in five internationals although Iordanescu would not have been pleased when Georgia substitute Levan Mcedlidze scored after 68 minutes following an error by Moti.
However, Romania's Gabriel Torje restored the home side's three-goal lead with an easy close-range finish in the 80th and fellow substitute Claudiu Keseru completed the rout with a deflected 25-metre shot four minutes from time.
After kicking off their Euro 2016 campaign against France in Paris, Romania will take on Switzerland five days later before meeting Albania in their final Group A game.
(Reporting by Angel Krasimirov; editing by Ken Ferris)
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