Mann's 10th goal of the season came with just under half an hour played as he hit a rasping left footed drive that beat 'keeper Nick Collick all ends up.
The opening fifteen minutes were fairly even, with Crowborough shading possession without fully threatening the United defence.
On twenty minutes, the returning Bailo Camara produced a superbly weighted through ball for Jason Taylor, who cut inside but saw his effort deflected over the bar.
From the resulting corner, the ball fell at the feet of Camara who lifted the ball over the bar from 10 yards.
Taylor was further denied at close range by Collick as United were in the ascendancy, and a prolonged spell of pressure resulted in Mann's goal.
Russ Tanner had to be alert to save from Ryan Waterman, holding the ball well in difficult conditions.
The second half started brighter for Crowborough as Wayne Clarke cut in from the right, but dragged his effort wide of Tanner's far post.
Aaron Capon, a marauding threat from right back all evening, superbly regained possession and played in Joe Summerbell who centred for Mann, but the United striker was unable to hit the target.
Mann saw another effort hit the far post before Camara saw a diving header go agonisingly wide.
Camara did see the ball go into the back of the net, but his effort was ruled out for offside.
Crowborough pushed forward in the final five minutes looking for an equaliser, but United withheld their pressure and progressed in Round Three