Farsley had taken a first-half lead through Adam Clayton but looked to be consigned to another year in the First Division North after Danny Frost's brace put Ossett 2-1 to the good.
However, Celtic scrambled home a dramatic 91st minute equaliser through Richard Marshall and took the final into two additional periods.
Neither side could break the deadlock in a tense opening half but with penalty kicks approaching, Ryan Watson slammed home from the edge of the box.
Two minutes later, Celtic sealed the victory as leading scorer James Walshaw curled home.
Farsley had been favourites to win the play-offs after narrowly missing out on the title to Lancaster City.
They will move into the seventh tier of English football.