James Dean added a third late in the second half, his fourth in four league starts since joining the club at the end of last month.
It was Morpeth who had the first clear-cut chance of the game, Sean Taylor firing the ball over the bar after a cutback from Liam Noble.
Brig’s Ali Waddecar was then brought down just outside the box as he attempted a driving run on 11 minutes.
Dudley placed the ball down and sent an inch-perfect free-kick over the wall and into the top left corner to give the hosts the lead.
Brig continued to press as the half went on and Kieran Charnock headed straight at goalkeeper Dan Lawson.
But Morpeth were still proving a threat, and Noble’s curling effort from 20 yards looked destined for the top corner but instead hit the underside of the bar.
Lloyd Rigby was then called into action to hold a Jeff Henderson long-distance strike.
Brig extended their lead 12 minutes into the second half.
Dudley was played clean through by a clever flick-on from Dean, before flicking the ball through the keeper’s legs and into an empty net.
Dean then came inches away from making it three – Dudley pulled it back towards the Brig number nine from the right-hand side, before Dean was denied by a fantastic stop from Lawson.
Dean was not to be denied a place on the scoresheet, however, again thanks to great build-up play from Joey Collins’ side.
It was Waddecar this time who showed good skill down the right to square to Dean, whose shot on target deflected off a defender and nestled into the bottom corner.