Moments after the break, Luke Summerfield fired in a free-kick for the visitors, but the League One side managed to hold on.
Visiting keeper Romain Larrieu was in the thick of the action early on denying Barcham and then pulling off a great save to keep out a poor clearance from his own defender David McNamee.
But the home side did make the break through when Jackson capitalised on some good build up play from Barcham and Curtis Weston.
Argyle were still recovering when moments later Barcham headed home debutant Chris Palmer's corner-kick at the far post.
The visitors did pull one back on 49 minutes courtesy of Summerfield's low freekick before striker Jamie Mackie wasted a glorious chance to send the game into the stoppage time right at the death.