Weston's double either side of the break clinched an emphatic win against struggling Argyle in the Bees' first game since Andy Scott was sacked.
The midfielder's first came 10 minutes before the interval when he arrowed home a far post header from Sam Saunders' precise cross.
And his second 10 minutes after half-time came at the end of a 30-yard run from Charlie MacDonald, who squared to Weston and he curled the ball into the top corner to give Plymouth goalkeeper David Button no chance.
Button kept his side in the game with a string of saves, keeping out headers from Karleigh Osborne and Craig Woodman.
The closest the Pilgrims came to breaking down a resolute Bees defence came after 12 minutes when Rory Fallon fired a fine 20-yard drive past keeper Richard Lee but against the crossbar.
Source: PA
Source: PA