Jon Ellis put Brady ahead from the penalty spot after Brixton skipper Weicz handled on the line. Two further Green goals wrapped up a great win.
Brady welcomed back Josh Gordon for his first game in six weeks, while they also had Dan Green making his debut.
Delighted Brady boss Joel Nathan told JC Sport: "Overall it was a very good performance in very tough conditions. Despite going two goals down early in the second half, I always thought we were always in control and once we finally found the net we never looked back.
"I would just like to say Ben Joseph was immense today and ran himself into the ground and showed what a great player he is. Also Rob Lerner tore them apart in the second half with some great play on the right-hand side.
"It is great to get a win under our belt after some disappointing results and now with some boys getting back to fitness I am sure we can go on a good run and show everyone in the league what we are about."
Brixton boss John Nitka told JC Sport: "We fell to pieces after a couple of contentious decisions."
