Ahmet Brkovic had given Town a first-half lead only for Paul Gallagher to level on the stroke of half-time after a moment of madness from rookie goalkeeper Dean Brill.
The Hatters pressed strongly after the break, but a frantic finale saw Gallagher put Stoke 2-1 up in the 83rd minute before Kevin Nicholls looked to have secured a point with a penalty three minutes from time.
But panic in the home defence led to Coyne's own goal late on giving the Potters the points without their eight-goal leading scorer Sam Bangoura.
A tentative start from both sides saw the first real opening fall to the visitors when Mamady Sidibe almost got on the end of a dangerous Karl Henry cross in the eighth minute only for the ball to drop to Brill.
Luton woke up and registered the game's first effort on target in the 14th minute when Nicholls fed Steve Howard but Rowan Vine's right-footed shot was saved by Steve Simonsen.
A poor Kenilworth Road playing surface was not helping the flow of the game and the uneven ground aided Luton in the 17th minute when Sidibe burst past Coyne down the right and crossed for strike partner Gallagher to mis-hit his shot from six yards into the chest of Markus Heikkinen, before Brill gathered.
Four minutes later the deadlock was broken as Edwards helped a ball over from the right for Howard to chest down and Brkovic rifled home a six-yard left-foot volley into the top corner for his fifth of the season.
The scrappy nature of the game continued until Luton went close again in the 39th minute when Howard nodded back a right-wing Edwards cross into the path of Steve Robinson who could only volley over from 14 yards.
Howard almost made it two three minutes before the break with a cheeky back-heel from Nicholls' run and cross on the right but Simonsen clawed the ball off the line.
Howard then struck a deflected effort into the arms of Simonsen from the edge of the area in the final minute of the first half, but Luton were shocked in stoppage time as Stoke grabbed an equaliser almost from nothing.
There didn't seem too much danger as the Potters launched a long ball in behind Kevin Foley, but as Brill came needlessly racing out of his box, Sidibe took advantage by nipping in, rounding the stranded keeper and crossing for Gallagher to tap into the gaping net.
The mistake seemed to unnerve the young Town stopper as just three minutes into the second period he dropped a speculative 25-yard effort from Stoke left-back Lewis Buxton only to gather at the second attempt.
Town pushed on with Edwards again looking dangerous down the right and from the Trinidad & Tobago international's 54th minute cross, Vine powered a 10-yard header into the arms of Simonsen.
Luton finally enjoyed a spell of possession which they almost made count when Nicholls' curling 20-yard free-kick was superbly tipped over by Simonsen in the 65th minute.
Howard hit a side-foot effort at Simonsen six minutes later, but Stoke caught Luton napping again seven minutes from time as Gallagher grabbed his second of the game.
With Heikkinen off the field receiving treatment, Sidibe crossed from the right for his strike partner to sweep home at the near post from eight yards.
With time running out, Henry brought down Howard in the penalty area in the 87th minute and Nicholls stepped up to blast home his fourth of the season.
A point looked secured until a mix up in the home defence in stoppage time led to Coyne lobbing the ball over Brill for an unfortunate own goal.