Dudley's first goal of the season came after Jason Gilchrist had opened the scoring for the visitors on five minutes, that being the first of a hat-trick of goals for the afternoon.
The South Shields number nine saw team-mate Darius Osei have an effort cleared off the line by Brig, before putting the ball over the line himself to give the visitors the lead.
Brig's goal came when Callum King-Harmes drove past two defenders before cutting the ball back to Dudley, who placed the ball into the corner of the net.
Danny Forbes produced a moment of skill when he nutmegged the opposing defender before his fierce effort was well saved by Myles Boney.
Shields then restored their lead at the other end, Gilchrist getting a touch to Osei's cross to beat Danny Roccia in the Brig goal.
Jamie Milligan's side then came close to restoring level terms in the second half, Macauley Wilson coming close with a free kick that was collected by Boney.
On 59 minutes a Brig corner was cleared as far as Matt Lawlor on the right flank, whose driven cross evaded everyone but was inches away from falling within the post.
Gilchrist then scored his third goal of the afternoon two minutes later, getting into position on the edge of the box and producing a right-footed finish.
As they had done throughout, Brig continued to work hard to get back in the contest and came close when Wilson's header looped onto the top of the net, but they were unable to make inroads into the scoreline and it was South Shields who took the three points.