The Shaymen had drawn seven and lost five league matches since beating Gateshead 1-0 on August 27.
It was not pretty for the most part, but points took precedence over performance given the winless run the Shaymen were on, and the overdue victory is just what they needed heading into the home FA Cup replay with Morecambe tomorrow.
Kosylo’s goal in the 39th minute was just what they needed after a scrappy display up to that point, as his shot from the edge of the area found the top left corner with the aid of a very welcome deflection.
Halifax should have doubled their lead when Cameron King’s deft touch played in Dayle Southwell at an angle but his venomous shot crashed off the bar.
Southwell nearly set up a second shortly after the interval with a terrific run and cross that was helped on by outstanding midfield man Cameron King to Kosylo, but his prod goalwards rolled just wide.
Southwell and Inih Effiong then traded chances, with both shots at an angle kept out, before Effiong miscued a shot wide from Dover’s next attack.
The Shaymen needed a second goal to calm the nerves, with Dover’s Stuart Lewis thundering a shot just over the bar from Sam Johnson’s punched clearance, before Effiong failed to convert from a couple more half-chances.
The Shay held its collective breath when Johnson kept out a low shot amid some late pressure from the bottom club.
FC Halifax Town: Johnson, Hanson, Brown, Clarke, Skarz, Maher, Staunton, Odelusi (Preston 61), King (McLeod 89), Kosylo, Southwell (Edwards 83). Unused substitutes: Rowley, Sellers.
Dover: Walker, Passley, Lokko, Brundle, Allen, Effiong, Gomis (Nortey 67), Lewis, Debayo, Taylor, Reason (Diarra 84). Unused substitutes: Adebowale, Worgan, Huckle.
Referee: R Welch.