In the opening stages, Alfie Usher stung the palms of goalkeeper Addison Bliss with a 20-yard free kick.
Josh Batty then had the ball in the net after latching onto Jack Richardson’s pass, but his effort was ruled out for offside.
Staveley grew into the game and took the lead through Luke Mitchell, whose firm drive from the right side of the box found the far corner of the goal.
The visitors almost doubled their lead towards the end of the half when a shot was blocked off the line by Finley O’Sullivan.
Although the Zebras were a goal down at the interval, their second-half performance was much better.
Brigg began the second half brightly showing more purpose in their attacks with Alfie Usher and Kevan Kahoussi looking sharp on either wing.
Alfie Usher equalized in the 56th minute after Dayle Hutson squared for the youngster to fire home from ten yards.
The Zebras began to control the game from then on with Hutson and Kahoussi always proving to be major threats.
Batty and Kahoussi both came close as Brigg looked for a second goal. It came late on in fortuitous circumstances.
In the 80th minute Josh Batty’s flick on from outside the box eluded Bliss and somehow ended up in the back of the net. Although Hutson stretched a leg out, which may have deceived the Trojans stopper, he didn’t quite get a touch on the ball so it was Batty’s goal.
Six minutes later it was game over when Dayle Hutson ran onto Finley O’Sullivan’s ball over the top and scored with a delightful chip, securing three points for the Zebras.
Brigg host Rossington Main in the NCEL League Cup on Wednesday evening.