Clovis Kamdjo put the Bees in front at half-time but substitute Dennis Oli, Obita and Charlie Lee hit back for the Gills to wrap up victory on the back of last week's memorable 5-4 triumph over Hereford.
It was a fourth consecutive win for Andy Hessenthaler's side and takes them to within a point of the play-off places, while struggling Barnet still have not won a league game since January.
There was a sour note for the hosts, however, as defender Garry Richards was stretchered off with a suspected broken leg.
Barnet took the lead six minutes before half-time. A cross from Ricky Holmes found Kamdjo, who guided the ball in at the near post. But half-time substitute Oli quickly made an impact two minutes after his introduction as he finished off a superb run down the right, burying the ball into the back of the net.
In the final 15 minutes, Gills striker Ashley Miller played a part in both goals. First his shot fell to Obita, who tapped home after Liam O'Brien's save, and then his neat pass with three minutes remaining allowed Lee put the gloss on the win.
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