The experienced goalkeeper dropped Peter Sweeney's 30-yard strike in the 69th minute to allow Luke Chadwick to level for the Potters, just three minutes after he had tipped Mamady Sidibe's overhead kick over the bar.
Stoke had taken the lead through Sweeney in the tenth minute, only for Town to turn things around with goals from Leon Barnett and Richard Langley at the start of the second-half.
But the hosts had to settle for a point after Beresford's blunder. It could even have got worse as Sidibe hit the crossbar with a header in the final minute.
In pouring rain at Kenilworth Road the visitors started the brighter with full-back Carl Hoefkens causing problems with some overlapping runs down the right.
It was Luton though who had the first attempt of the game in the fourth minute when Langley collected Kevin Foley's pass and hit a 25-yard effort that flew four yards over the crossbar.
Stoke continued to dominate the midfield area and went ahead with their first real effort in the tenth minute.
Vincent Pericard was first to a long free-kick and knocked the ball back into the path of Sweeney, who hit the ball first time from the edge of the area and was perhaps a little surprised to see his effort somehow find its way into the bottom right-hand corner of Beresford's net.
That setback shocked Town in to action and Carlos Edwards, once again looking lively for Luton, took a pass from Peter Holmes five minutes later, cut inside his marker and unleashed a fierce 22-yard effort at Steve Simonsen.
Foley was next to try his luck as he advanced from the back in the 21st minute, played a neat one-two with Warren Feeney and surged into the penalty area only to be denied by a well-timed challenge from Clint Hill.
Lewis Emanuel twice went close before Rowan Vine saw a long-range effort saved by Simonsen as Luton searched for an equaliser.
After spending most of the first-half defending their lead, Stoke mustered an attack three minutes into the second period when Chadwick was first on to a long Simonsen clearance and laid the ball off to Darel Russell to fire just wide from 25 yards.
The familiar pattern of the game soon returned, however, as Coyne came agonisingly close to a leveller a minute later when his diving header from Langley's corner came back off the crossbar.
The goal the hosts deserved finally arrived in the 54th minute when Barnett rose to flick the ball home from an Emanuel cross after City had struggled to deal with a Langley corner.
Town then moved ahead on the hour after getting the better of two penalty claims that came in the space of three minutes.
Firstly Pericard tumbled in the area under Barnett's challenge only for referee Gary Sutton to ignore the visitors' appeals.
But then when Morgan went sprawling under Michael Duberry's challenge, Sutton pointed to the spot and Langley, who had had a poor game up until that point, found the top corner of the net from the spot.
Stoke rallied and levelled in the 69th minute after Town keeper Beresford experienced contrasting emotions in the space of four minutes.
The veteran shot-stopper had done superbly well to tip giant Potters striker Sidibe's spectacular overhead effort over the bar in the 66th minute.
The same could not be said of his handling of Sweeney's speculative 25-yard shot minutes later as he let the ball slip from his grasp for Chadwick to fire into the roof of the net from six yards.
Emanuel then fired over after Vine's pass in the 74th minute, Vine himself had an overhead effort saved in the 78th minute and Holmes shot wide two minutes from time, but Luton couldn't find a winner.