Kosylo’s brace and debutant Connor Thomson’s late third extended the home side’s winless run in the National League to five games.
The visitors took the lead from the penalty spot when Kosylo was taken down in the area by Kane Ferdinand and stepped up to send goalkeeper Nathan Baxter the wrong way.
Woking levelled soon after half-time when Charlie Carter diverted team-mate Regan Charles-Cook’s shot into the net.
But Kosylo put Halifax back ahead when he found the far top corner from a tight angle, and Thomson wrapped up the points by turning in Michael Duckworth’s cross.
In National League North, Harrogate Town thrashed FC United of Manchester 6-0.
There were six goals too in Bradford Park Avenue’s game with Alfreton Town, but they were shared in a 3-3 draw.
York City beat Boston 1-0, while North Ferriby United earned a 2-2 draw at Nuneaton.