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 on 7 minutes, Sean Taylor spooning the ball over the bar after a clever cut back from Liam Noble in the box.
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 Brig the lead.
Brig continued to press as the half went on and Kieran Charnock headed straight at goalkeeper Dan Lawson after meeting another Dudley set piece delivery.
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 then extended their advantage 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 &placing the ball into an empty net on the slide.
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, who got across quickly to deny him from close range.
Jack Foalle then worked an opportunity down the other end for Morpeth, his effort on the half volley after good control subsequently going over the bar.
Dean was not to be denied a place on the scoresheet, however, again thanks to great build up play from Joey Collins's side.
It was Waddecar this time who showed good skill down the right side to square to Dean, whose shot on target deflected off a defender and nestled into the bottom corner.
That was the last real talking point from the game, as Brig saw out the remaining time to record a deserved win - their second home win in three games, with a chance to make it three out of four on Tuesday night against Nelson in the LFA Trophy.