Lewis Maloney shot straight at the keeper in Boro's first attack, then ex Boro centre-back Shane Killock fired over for Stalybridge, writes Steve Adamson.
Boro soon got on top, creating numerous chances, with a Jake Day shot blocked, and Bailey Gooda having a header from an Ash Jackson cross cleared off the line with the keeper beaten, then Cameron Wilson shot wide of the left-hand post, and Maloney fired just over.
Kieran Glynn was on top form, and he dinked the ball forward to Plant, whose first time shot fizzed inches wide.
Simon Heslop and Wilson both had efforts stopped by home keeper Liam Hughes, and a Brad Plant shot struck the left-hand post as the pressure continued, but Celtic then created a chance, as Ntumba Massanka headed on to Raul Correia, but Will Thornton got in a great tackle as he was about to shoot. Boro were dominant as half time approached, and a Maloney free-kick was headed across the penalty area by Thornton, towards Plant, who swivelled to fire his shot inches wide.
Heslop and Glynn were bossing midfield, and the former played a ball forward to Plant, who flicked on to Maloney, who dragged his shot past the post, then Plant had a shot comfortably saved, and shortly before the interval, Thornton appeared to be kicked from behind in the Celtic area, but the referee waved away penalty appeals.
Boro totally dominated the first half, but the home side were more competitive after the break, without ever troubling Ryan Whitley in the Boro goal.
Kieran Weledji and Ash Jackson both worked hard, overlapping down the flanks, and Gooda and Thornton were outstanding at the back, winning headers and tackles to break up several Celtic attacks.
A Maloney cross was headed narrowly over by Day, and great interplay between Glynn and Plant set up Day, who fired over, while for the home side, Sam Wedgbury blasted a free-kick just wide.
Maloney dinked a shot past the far post, then had another shot headed clear by Killock.
With time running out, sub Luca Colville beat two defenders before having a fierce shot deflected by a defender, and Heslop laid off to sub Ryan Watson who also had his shot blocked by the hard working home defence.
Then a throw-in from Jackson fed Maloney on the left, and he sent over a cross cum shot towards the six-yard box, which Gooda tried to get his head to, but the ball evaded him and ended in the net for a sensational winning goal, much to the delight of the 150 vociferous supporters.
Boro then held comfortably on for the remaining three minutes of stoppage time to claim a magnificent win, which takes them to within a point of the play-offs.
BORO: Whitley, Weledji, Jackson, Heslop, Thornton, Gooda, Plant (Watson 70), Maloney, Day (Colville 58), Glynn, Wilson (Cadman 80)
MAN OF MATCH: Bailey Gooda.