Defenders Tom Flanagan and Matt Fish grabbed their first goals of the season to put the Gills in control of the game at half-time.
Zander Diamond pulled one back for the visitors, who arrived at Priestfield on an eight-game unbeaten run, before Myles Weston and Joe Martin struck in quick succession to put the result beyond doubt.
Gills were fastest out of the traps and went ahead when Flanagan capitalised on Chris Whelpdale's quick throw-in to fire home from close range after 22 minutes.
Weston and Charlie Allen came close to doubling the host's advantage before right-back Fish did so from 25 yards after Burton failed to clear their lines.
Diamond headed against a post at the start of the second half but he gave the visitors hope when he capitalised on some sloppy Gillingham defending to slot home after 54 minutes. However, any chance of a comeback was put to bed five minutes later when Weston latched onto Danny Kedwell's knock down before firing past Stuart Tomlinson. Martin then sealed all three points for the hosts with a spectacular half volley.