Prolific non-league marksman Mark Bellingham gave Daventry a deserved lead five minutes into the second half, but Slough equalised with a Dan Burnell penalty before Coyne's late strike snatched victory and capped a fine individual performance.
The visitors would have felt aggrieved with the result, as they squandered good chances before the break. Daventry dominated the early exchanges and played some neat attacking football, which had Slough on the back foot.
In the first minute, Stuart Swift got back well to cover and prevent Tom Burn from converting at the far post as Scott Cross played the ball into the six yard box.
Wide players Burn and Josh O'Grady, who posed problems galore for the Slough defence, fired efforts well wide as Daventry kept the pressure on, but their best chance came from a corner. Craig Milligan headed against a post, and from the rebound, James Warrington was beaten by a fierce strike, but Danny Murphy was well placed to clear off the line.
Good build up play saw Coyne chip one over the top for Burnell, but he could not get a touch to divert it past keeper David Bevan, and that was as good as it got in a frustrating first half of few chances for Slough, who also lost Steve Sinclair with a hamstring strain, to be replaced by Shaun Byrne.
After a decent start to the second half from both sides, the visitors took the lead with 51 minutes played. Slough lost the ball in midfield after attacking from a corner, and Burn broke away at pace. He played in Nathan Hicks, who burst down the left side of the box and squared the ball across for Bellingham, whose left footed effort beat Warrington's despairing dive.
Within five minutes though, Slough were level. Referee Oldershaw penalised the visitors for some holding in the penalty areas as a free kick was swung in. Dan Burnell stepped up confidently and sent David Bevan the wrong way with a low placed effort.
Bellingham bent a free kick just wide, but Slough grew in confidence and went on to dominate the rest of the half. Burnell turned wonderfully and fired a cross-shot across the face of goal, but just too far ahead of the stretching Sean Sonner.
Burnell himself went agonisingly close to a winning goal as he failed to make contact on Murphy's corner from close range. But at the other end, Luke Edwards headed a corner powerfully over after he broke clear of his marker.
Burnell was proving more of a menace as the game drew to a close, heading one effort straight at Bevan and testing the keeper again with a low strike after good work from substitute Ben Abbey.
Slough's moment was to come, however, as a reward for pressing on after grabbing the equaliser. Stuart Swift broke down the left and played the ball across for man-of-the-match Paul Coyne to flick in from three yards.
Daventry Town Line up
David Bevan, Lewis Bishop, Craig Milligan, Peter Saunders, Aaron Parkinson (c), Luke Edwards, Josh O'Grady (Claudiu Hoban), Nathan Hicks, Scott Cross (Jason Simpson), Mark Bellingham, Scott Burn (Adam Confue). Subs not used: Jason Lee, Howard Forinton.