The win sees the Capital outfit leapfrog Rangers B into second spot in the Lowland League table.
Dishington's last-gasp goal was no more than the home side merited following an excellent second-half display in front of a crowd of 317.
Dougie Samuel made four changes to the side that needed a last-gasp penalty to salvage a 1-1 draw against Stirling University on Saturday. Spartans stalwart Blair Carswell replaced Mikolaj Komocki between the sticks with a trialist, Sam Jones and Sean Brown also given the nod from the start. Former Edinburgh City striker Blair Henderson dropped to the bench while Scott Maxwell and Finn Robson missed out altogether.
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The visitors broke at a frantic pace in the fifth minute, but winger Ben Summers’ effort drifted wide. Sixty seconds later and it was the home side's Cameron Russell who forced a fingertip stop from Ryan Mullen to prevent an early breakthrough.
Celtic' s Ben Wylie's incisive lob over the top then sent Owen Moffat scampering towards the by-line. The winger managed to hook the perfect ball back across goal only for the chance to go abegging.
However, the Glasgow outfit did edge in front in the 27th minute and what a classy finish it proved. Joey Dawson collected a pass inside the penalty area, rolled his marker and expertly finished into the bottom corner.
Celtic had found their mojo and Wylie was next to send an effort hurtling towards Carswell's goal, the ball just a yard or two over the crossbar.
The home side almost drew themselves level when Euan Smith's header back across goal set off panic in the Celtic rearguard before Mullen was finally able to gather.
Then, just 60 seconds prior to the break, Dishington's in-swinging corner almost caught out Mullen as the visiting No.1 had to divert the ball over his own crossbar. But it was second time lucky for Dishington as his deep ball to the back post was met by Kevin Haugh, who sent a downwards header into the net for the equaliser.
Carswell was brave right at the start of the second half to deny Moffat from point-blank range and Sean Brown was putting the visitors' defence under considerable pressure with his willingness to chase down every ball.
But Celtic spurned a prime opportunity to restore their lead on the hour mark when Summers appeared to get the ball stuck between his feet at the back post.
Dishington cracked the outside of the post when he should have done better as Spartans really began to impose themselves and the same player dragged a shot just wide of the post.
The hosts the woodwork for a second time with ten to play as Russell's corner hit the top of the crossbar. But Dishington finally made it stick when he turned to fire home in the final minute following substitute Henderson's flick-on.
Spartans: Carswell, Smith, Allan, Waugh, Trialist (Herd 88), Woods, Dishington, McFarland, Brown (Henderson 79), Russell, Jones. Subs: Walker, Newman, Inglis, Komocki.Celtic B: Mullen, Luyeye, Paterson, Easton, Corr, Murphy, Moffat, Wylie, Dawson, Letsosa (Deane 72), Summers. Subs: Robertson, Morrison, Dobbie, Kelly.