Elijah Adebayo set the Town on their way with his third goal of the season, heading home a cross from strike partner Carlton Morris in the 18th minute.
A 77th-minute second, which was originally credited as Jimmy Dunne own goal but owed everything to sub Cameron Jerome's persistence and replays seemed to back up the striker's claim that it came off his knee last, looked to have secured the win.
But an unfortunate 91st-minute Ethan Horvath OG at the other end gave Rangers hope, only for Luke Freeman to grab one against his former club at the death to seal the three points.
Manager Nathan Jones named an unchanged starting line-up from the goalless draw at West Brom, with Sonny Bradley back leading the side for the second match in a row following his clean sheet against West Bromwich Albion last weekend.
Rangers, third on goal difference but level on points with top-of-the-table Sheffield United and second-placed Norwich City at kick-off, started on the front foot and Tom Lockyer had to produce a perfectly timed tackle in the box to deny Luke Amos a shooting opportunity after three minutes.
But Town were soon on the attack with Adebayo seeing an angled shot deflected wide for one of a flurry of Hatters corners in the opening exchanges, before Henri Lansbury rolled the ball into James Bree's path on 11 minutes, but the right wing-back's 20-yarder was always rising.
The breakthrough wasn't long in coming, however, and it was that combination up front again who proivided it, Morris sending in a delicious cross from the left byline that Adebayo gobbled up with a nod of the head from close range for his third goal in as many matches.
His brace against Huddersfield in the last home match both came courtesy of assists from Morris, and the six-goal top scorer was once more the architect here as the front pair's partnership continued to flourish.
The visitors looked to hit back and Lansbury timed a sliding tackle brilliantly on the edge of the box just as Rangers captain Ilias Chair prepared to pull the trigger, then Lydon Dykes headed a Jake Clarke-Salter cross straight at Horvath before the back three of Lockyer, Bradley and Dan Potts all got their heads on crosses to clear danger as Town stood strong.
The Hatters were playing well in a cracking game, however, and after some clever play on both sides of the penalty area, Lansbury sliced a half-volley wide that looked perfectly set for him, when considering the one he struck from 25 yards out at Hull a fortnight ago.
Town were knocking at the door and Bradley saw a header cleared off the line before Bree, also the provider of that chance, whipped two more balls in from the right; the first saw a Morris shot blocked from a clever free-kick routine, before Adebayo's header thumped the inside of the post and went across the goalline with Hoops keeper Seny Dieng pawing the ball away.
Jordan Clark became the first player from either side to be shown a yellow card for a late challenge on Tim Iroegbunam a minute before half-time, but Horvath was right behind Chair's 30-yard free-kick and the Hatters went in at the break holding the advantage.
Then second half started in similar fashion to the first, Lockyer blocking a shot from Scotland striker Dykes on the edge of the area
Then, with 52 minutes on the clock, Morris and Adebayo combined on the right to set Clark racing into the QPR box, only for a sliding challenge from substitute Osman Kakay to deflect the ball wide for a corner as Dieng spread himself.
Dunne fired over from 20 yards at the other end as Rangers threatened, then Tyler Roberts hit a left-footer from further out that fizzed just past Horvath's left post.
Jones made a triple change midway through the half, with Harry Cornick, Jerome and Freeman coming on in place of Morris, Adebayo and Lansbury, who was by this point on a yellow.
Town survived a huge scare on 73 minutes when Roberts smashed a shot against the crossbar, with Horvath reacting well to fall on Clarke-Salter's follow-up effort.
Campbell came close to extending the Hatters' lead in the 76th minute when Dieng had to back-pedal to tip Campbell's attempted lob over the bar, then Cornick sliced wide from the resulting corner.
Within a minute the lead was doubled and it owed everything to Jerome's persistence, first in nicking the ball ahead of Dunne, who then had the ball smashed against him by his own keeper Dieng before Jerome shepherded it over the line, convinced he had touched the ball last, although it was officially credited as an own goal at the time.
It would be an unhappy end to Dunne's afternoon, as the central defender had to go off injured courtesy of Dieng's follow through.
Jerome raced clear again moments later, this time crossing for Clark to steer wide when Freeman might have been better placed to take it on his left.
Town survived an 87th-minute scare when a free-kick from wide on the left by Chair was deflected through the six-yard box, but Rangers did pull one back when Andre Dozzell's cross skimmed off Bradley's head, came down off the bar and went over the line via Horvath's back.
There was still time for Luton to restore the two-goal advantage, however, when Jerome and Cornick combined intrucately on the right edge of the box for the latter to square for Freeman to flick home his second goal for the Hatters three minutes into injury-time.
A first win over Rangers in seven attempts since returning to the Championship took the Hatters up to sixth before the afternoon and evening kick-offs, and on a six-game unbeaten run heading into a midweek trip to face another top side in Norwich.
Goals:
LT - Adebayo 18, Jerome 77, Freeman 90+3
QPR - Horvath (OG) 90+1
Att: 10,011 (1,038 away)
