The home side could have taken the lead after five minutes when Alex Meechan latched on to Ben Cleverley's long ball forward, but his shot was saved by Tony Pennock.
Six minutes later the deadlock was broken when Cleverley's inswinging corner was not cleared by the visiting defence and in the goalmouth scramble Jon Richardson forced the ball into the back of the net.
In the 16th minute the lead was increased when Mathew Russell's deep cross fell to Kayode Odejayi to fire the ball home with outside of his foot.
Farnborough's first chance of the game came in the 20th minute when a long ball over the top of the home defence found Paul Harkness, but his drive was comfortable for Steve Perrin to block.
The game was sewn up seven minutes later when Meechan shrugged off his marker on the right and sent in a pin-point cross which Odejayi converted for his second and Rovers' third.
Meechan was unlucky not to get in on the act just before the half hour when he went on a surging run only for his drive to go inches over the crossbar.
The visitors were unlucky not to grab a goal back after 33 minutes when Geoff Pitcher's free-kick left Perrin beaten, but the ball rebounded off the post before being cleared to safety.
The easiest chance of the half came Rovers way Odejayi sent Neil Grayson clean through, but with just the keeper to beat he fired wide of the left post.
Despite dominating the first half Rovers took their feet off the accelerator and allowed Boro to get back into the game with Steve Jenkins' mistake allowing Rocky Baptiste to square the ball for Harkness and he finished comfortably just after the hour.
Buoyed by the goal, Farnborough continued to pressure the Forest Green defence but they and Perrin managed to keep them at bay.