Ian Harte replied for the visitors with a hotly-disputed penalty after Brentford defender Leon Legge challenged the onrushing Scott Dobie in the box.
Harte's spot-kick made it 2-1 with just 10 minutes remaining but Andy Scott's side sealed the win when Dickson hammered home from close range in the 93rd minute.
Weston, who scored twice against the Cumbrians on the opening day of the season, put the Bees ahead on the half-hour when he side-footed home following a jinking run from Charlie MacDonald.
He made it two just after the break, drilling home from the edge of the area after John Bostock's shot had been well saved by Adam Collin.
Source: Team Talk
Source: Team Talk