AFC boss Martin Baird fielded a strong side with Jake Darnell set to make his return, only to then pull up during the warm-up.
It proved a blessing in disguise as Clarke, on loan from Fleetwood Town, capitalised on AFC’s first opportunity of the game 10 minutes in.
A well organised corner routine saw Adam Sumner find the attacking midfielder at the back post to nod home on his debut.
Ben Duffield tapped wide from Gregory’s cross and Josh Winder saw an effort tipped behind before AFC were pegged back in the 31st minute.
It came in a similar fashion to Clarke’s goal with Jack Hunter found at the far post, where he nodded home to equalise.
AFC responded as Clarke found the net again just before half-time, only for his header to be ruled out for offside.
Having missed chances in the first half, the Mechanics needed to come out stronger after the break if they wanted to claim the points.
They regained the lead within eight minutes of the restart when Joe Robinson played the ball out to Ryan McLean on the left-hand side.
He took one touch before looking up and spotting Gregory, running towards the front post.
He delivered a low cross and the forward tapped it in from close range for his second goal in as many games.
Winder was then thwarted by another offside decision and, with the game still in the balance, Celtic pushed for another equaliser but Baird’s side stood firm.
The win moved them into fourth place in the NWCFL First Division North table ahead of Monday night’s trip to Daisy Hill (7.45pm).
AFC Blackpool: Eastham, Radcliffe, Clarke, Bishop, McLean, Sumner, Thompson, Gregory, Robinson, Winder, Duffield. Subs: Hall, Hughes, Roberts, Smith, Darnell.
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