The table toppers raced into a two-goal lead inside 11 minutes, courtesy firstly of Jack Payne's 20-yard strike before Danny Kedwell nodded his fourth league goal of the season.
The Bees reduced the arrears when Jack Saville headed home, but Deon Burton restored the visitors' two-goal cushion six minutes before the interval.
After a quiet start, highly-rated 20-year-old midfielder Payne collected from a long throw before drilling a pin-point effort past a motionless Graham Stack.
And almost straight from kick-off it was two-nil as winger Charlie Allen sent over a delightful centre and an unmarked Kedwell rose highest in the six-yard box.
The Gills' quick-fire double might have prompted a procession, but the Bees halved the deficit via the head of Saville, as he expertly met Mark Byrne's 23rd-minute corner. But Allen's men extinguished any hopes of a revival when Kedwell's flick found Burton at the far post and he nodded past Stack.