The 26-year-old former Chester star scored the equaliser and created the winner for Dave Hibbert after Paul Mullin had put Stanley one up in the 39th minute.
Davies curled home a stunning shot into the top right-hand corner in the 64th minute, before he found Hibbert with a pin-point right-wing delivery just four minutes later. The 21-year-old powered a header home from inside the six-yard box.
It truly was a remarkable turnaround for the visitors, who were given a half-time rollicking by their manager Gary Peters.
They looked lethargic and, according to Peters, didn't deserve anything on their first-half showing.
Stanley should have been out of sight after Mullin headed home a John Miles corner six minutes before the break.
But they were made to pay yet again at home - they have only won four at the Fraser Eagle Stadium - by a Davies inspired Town.
After Hibbert had headed his ninth of the campaign, Town also had a third goal chalked off when Marc Pugh was adjudged to have carried the ball out of play during a mazy run.
Stanley then piled on the pressure in the last ten minutes, with Robbie Williams going the closest when he fired wide in a goalmouth scramble.
Stanley's boss John Coleman was absolutely livid with his side for not killing off the game, calling it "ten minutes of madness" that cost his side the contest.