Dudley scored either side of half time, before Marshall Willock halved the deficit on 60 minutes for the visitors.
The victory, Brig’s third in a row at home, moves them 12 points clear of the bottom two with 30 games played.
The game started at a high tempo and Grantham’s Jack Broadhead fired just wide of Danny Roccia’s post inside the opening minute of the game.
Brig then took the lead two minutes later, Rowan Roache finding Dudley with an exquisite outside of the foot pass over the top of the defence.
Dudley made no mistake, beating the offside trip and finishing calmly past Grantham stopper Jon Worsnop into the bottom corner.
Both sides then settled into a rhythm and it took until 23 minutes before the visitors registered their first effort on target, Craig Westcarr miscuing a shot straight into the hands of Roccia.
Brig then had strong claims for a penalty turned down when Dudley was brought down inside the box, before Roccia made a superb stop down the other end to deny Akeel Francis’s goalbound header.
Jamie Milligan’s side started the second half strongly and were rewarded with their second goal of the afternoon.
George Thomason, making his first Brig start following a loan move from Bolton, beat several defenders on a driving run before squaring to Roache on the right flank.
Brig’s number seven’s pinpoint cross picked out the onrushing Dudley, who tapped home his second goal of the afternoon.
From then on they had to deal with increased pressure from the Grantham attack, Roccia making another top save to deny Curtis Morrison’s curling effort.
From the resultant corner, the ball found its way to Willock, who poked the ball over the line from six yards to give the visitors hope.
Chris Robertson then headed narrowly over from another corner, before the returning Ali Waddecar was unable to convert from a tight angle on the counter attack for Brig.
Roache then jinxed his way through the Grantham backline before seeing his shot denied by a good low save from the ‘keeper, and Brig held out during a passage of sustained pressure late on at the other end to secure a vital three points.