Despite Brig have more of the ball and looking more threatening in advanced areas, neither goalkeeper was tested in the first half as the two sides went into the break with the scores level.
Callum King-Harmes came closest for Brig as he fired an effort narrowly over the bar, before Isaac Sinclair closed down Matlock stopper Shaun Rowley's clearance and almost inadvertently got on the scoresheet himself.
The game began to open up more in Brig's favour in the second, with Richie Allen driving past the defence and forcing a good save out of Rowley.
The Town goalkeeper was called into action again to turn Isaac Sinclair's close range effort around the post.
The following corner initially came to nothing for Brig as Matlock cleared, but Allen rescued the situation by controlling the ball on the left with his back to goal, before firing the ball into the box.
There Burton was waiting, calmly volleying the ball over the head of the stranded Rowley and into the corner of the net.
Matt Dudley then came close to adding a second when his effort was saved, before Danny Forbes almost scored a goal of the season contender when his dipping volley from about 50 yards out hit the stancheon behind the goal.
With five minutes additional time, Matlock piled the pressure on, and Danny Roccia had to be alert to tip Andrew Bond's effort over the bar.
Brig were able to clear the subsequent corner and see the match out, meaning Jamie Milligan's men secured an excellent opening day victory over a side that won by a reverse scoreline against Brig at the same point last season.