Barcham slammed in from 25 yards in the opening six minutes of Saturday's League One clash at Priestfield and Weston matched his moment of magic before half-time as the Lions struggled to cope with wave after wave of home attacks.
Barcham broke the deadlock when he fired a rocket beyond David Forde and into the top corner.
Simeon Jackson thought he had doubled the lead after 25 minutes when he nipped past Andy Frampton and fired in a close-range volley but his effort was ruled out for offside.
But the hosts got their second five minutes before the break when Weston capped a perfect first half for Mark Stimson's men with a powerfully struck 20-yarder.
A minute later Neil Harris wasted a great chance at the other end with a low shot against Simon Royce as the Lions looked for a way back into the game but the damage had already been done.