The visitors had the better of a dire first half but failed to make the most of a number of early opportunities.
Rangers had not had a single effort on goal up until the 41st minute, but after a messy goalmouth scramble, Dexter Blackstock side-footed home for his 12th of the season to put them in the lead.
But it was justice done when Luton went into the break on level terms with a goal on the stroke of half-time.
David Bell swung in a teasing left-foot cross from the right-hand side and Chris Coyne got the faintest of touches to take the ball past Lee Camp and into the net.
The goal came at the perfect time and gave Luton the belief they needed to go on and take a deserved lead six minutes after the interval.
It came via a penalty kick from Bell after Rangers striker Marc Nygaard had inexplicably used his hand to defend a Dean Morgan cross.
Rangers were left furious on 67 minutes when substitute striker Furlong appeared to be tripped in the penalty area by Markus Heikkinen and referee Clive Penton failed to award to spot-kick.
But the tables were turned on Luton when Furlong was brought down by the same man 11 minutes from time and Penton showed no hesitation in awarding a penalty which Blackstock converted.
It knocked the stuffing out of Kevin Blackwell's side and Furlong won the game for Rangers in the dying seconds when he headed home a Steve Lomas cross.