Davidson's goals came in a five minute spell as the Shaymen opened up a 4-2 lead at Dimple Wells Road.
And his second proved to be the winner after Ossett Albion pulled a late one back and would have forced the game into extra time but for a dramatic last save from Phil Senior.
Town made nine changes to their starting line up with only with only two players - Lincoln Adams and Steve Payne - remaining from the 7-1 win over Salford at the weekend.
But they were still too strong for an Ossett side that until Saturday had topped the Unibond League Division One North table.
The game started badly for the Shaymen as they went behind after just six minutes from Ossett's first corner.
Former Town man Michael Senior took the kick which namesake Phil was unable to get to and the ball went in off the post.
Andrew Seed fired wide soon after as the home side looked to press home their early advantage,
But Halifax started to get their act together with Danny Ellis prominent on the right.
Craig Smith's free kick was flicked on by Steve Payne, but too far in front of Lincoln Adams while Ellis fired across goal after Ryan Sugden had been fouled on the edge of the area.
In between, a Michael Senior free kick was claimed by Phil Senior.
And Seed had an effort blocked, but in the main it was Halifax who were on the front foot and Adam Morning got on the end of Nigel Jemson's 23rd minute flick only to be denied by Dominic Riordan.
Jemson flicked into the side netting then both he and Sugden missed an Ellis corner before Town levelled 10 minutes before the break.
Morning raced clear down the left, leaving the full back for dead, before whipping in a cross for Ellis who steered it home from close range.
The Shaymen finished the first half strongly with Kyle Cook forced to clear under pressure from Ellis and Sugden while Ross Wilkinson was forced to put a Lincoln Adams effort behind for a corner.
Michael Senior had an effort saved early in the second half.
At the other end, Jemson's downward header from a Morning cross was cleared but the duo combined again just before the hour mark as the Shaymen hit the front for the first time.
Scott Phelan's cross field pass initially picked out Ellis who returned the ball to his fellow newcomer.
Morning then sent in a low cross which, after home keeper Neil Bennett had parried, was turned in by Jemson.
Ellis's pass soon after was just too strong for Davidson to capitalise on and the substitute's shot on the turn from a Payne throw-in thumped against the underside of the bar and rebounded to safety as far as the home side were concerned.
Town eventually increased their lead in the 70th minute and Morning was involved again, providing the cross from the right which was volleyed home at the near post by Davidson.
Ossett hit back within two minutes when Seed got away down the left and crossed deep for Andrew Catton who headed home from eight yards.
Town were two goals in front again three minutes later after more good work by Ellis down the right. He skipped away from a couple of clumsy-looking challenges and crossed for Davidson who turned well to fire home.
Albion hit back again with eight minutes remaining, Jason Bentley curling the ball into the net after a good run and neat turn on the left.
And the home side threw the kitchen sink at the Shaymen in the closing stages, with Town indebted to Senior for two great saves.
The first from Bentley was low down at his right hand post from a Sam Jackson cross and his second was from a Catton corner that threatened to sneak into the net.
In between, Adams did well with a well-timed sliding challenge to deny Bentley a shooting opportunity.

MATCH FACTS
Ossett Albion: Bennett, Jackson, Cook, Allinson, Riordan (Grace 81), Wilkinson, Catton, Senior (Syers 64), Dickinson (Bentley 70), Kelsey, Seed. Subs (not used): Parker, White.
Booked: Kelsey (foul 90)
Sent-off: None
Goals: Senior 6, Catton 72, Bentley 82

FC Halifax Town: Senior, Hinsley, C Smith, Payne, Adams, Morning (Moore 87), Hilton (M Smith 60), Phelan, Ellis, Sugden (Davidson 60), Jemson. Subs (not used): Kennedy, King.
Booked: C Smith (foul 9), Phelan (foul 85)
Sent-off: None
Goals: Ellis 35, Jemson 59, Davidson 70, 75

Attendance: 248
Referee: N Bannister.