The Shaymen failed to build on an encouraging first-half by conceding twice after the restart through Tyler Cordner and Alfie Rutherford.
Niall Maher got one back for Halifax, but it’s now five defeats and seven draws in the league for Town since that win over Gateshead on August 27.
Halifax came the closest to opening the scoring in the first-half.
A Cristano Ronaldo-esque free-kick by Town’s number seven was brilliantly tipped behind by Havant’s player of the month Ben Dudzinski, before Josh Staunton headed over the resulting corner.
Another Kosylo free-kick was held by Dudzinski shortly afterwards.
From a Kosylo corner, Brown headed the ball back across goal where it was cleared off the line by Will Wood.
Winger Bradley Tarbuck fired just over from an angle inside the area for Havant, who produced some decent build-up play, but failed to register a shot on target in the first-half.
But Havant took the lead with their first effort on target just after half-time when a corner was headed high into the air and across goal by Cordner, and dropped down just beneath the crossbar, with Sam Johnson unable to claw it out.
And they were in real trouble when Rutherford tapped-in the influential Tarbuck’s low cross from the right, as Havant found it too easy to carve a route to goal.
Out of nothing, Maher halved the deficit when his innocuous-looking shot from 25 yards was spilled into the net by Dadzinski.
Town’s tails were up after that to set-up a tense climax, in which Will Wood was sent-off for what appeared to be a shove on a Halifax player.
Elsewhere, Aaron Williams’ goal deep into stoppage-time denied Chesterfield their first league win since August as Harrogate forced a draw.
Jonathan Smith put Chesterfield ahead when he tapped in a cross from second-half substitute Jerome Binnom-Williams.
Chesterfield almost went 2-0 up when Binnom-Williams had a late chance, but the shot was deflected behind for a corner.
Liam Kitching and Mark Beck both had headers saved with minutes to spare, but Harrogate got their goal in the seventh minute of stoppage-time when a corner fell to Williams.