The three points saw Brig move to within two points of fifth-placed Warrington Town – who were beaten 1-0 at home by Mickleover.
Town do have a game in hand but with four games of Brig’s season remaining, they are right in the mix to finish inside the top five.
A goal in each half was sufficient for Jamie Milligan’s men to secure the victory over Witton.
It was Joe Strawn who almost gave Brig an early lead but his superb volley crashed against the crossbar.
On 22 minutes, Paul Dawson curled an effort over the bar after good work by Aaron Skinner, Kevin Spinelli and Dan Martin.
Witton’s long throws were causing some concern for Brig but they took the lead on the stroke of half-time courtesy of a fine goal.
Nathan Pond’s towering header was superbly controlled by Mark Yeates who knocked the ball into the path of Dawson.
The former Watford man carried his run on and was brilliantly played in by Dawson’s reverse ball.
With just the keeper to beat, he toe-poked the ball home.
The opening few minutes of the second period saw Witton push the hosts back but just when they were getting a strong foothold in the game, Brig got the all-important second goal on the hour mark
Once again it was the Yeates-Dawson show, but this time it was the former’s turn to pick out the latter whose first-time shot sliced into the bottom corner.
Isaac Sinclair then saw his curling effort saved by Hall and with 10 minutes to go, the same player had a great chance to seal the win but he lifted the ball over the top when clean through.
Milligan said: "I thought the boys were really good today and deserved the victory. If you would have offered me seven points out of nine in the week going into last Saturday, I would have absolutely taken it.
“There’s no pressure on us, we’ve over-achieved all season, but we have four big games to go and have given ourselves a real chance. It’s going to be an exciting run-in and we’ll give it our all, starting at Hyde next week.”