The visitors opened the scoring through goal machine, Chelsea loan man Jimmy Smith, before the Hatters scored twice in first-half stoppage time through Adam Boyd and Ahmet Brkovic.
The strikes marked the first time Luton had hit the net in more than 400 minutes of football.
But Town couldn't make the most of that fact as Markus Heikkinen inexplicably headed into his own net moments before Dexter Blackstock swept home what proved to be the winner after more poor home defending.
The visitors started at breakneck pace and a flying 40-yard run from Lee Cook drew a foul from Heikkinen inside the first minute, but the winger could only pick himself up to curl harmlessly over from 20-yards.
Rangers kept pressing Hatters and created another half-opening when Cook crossed from the left for Marc Nygaard to turn and shoot wide from 12 yards.
Luton gradually got a grip on proceedings and came close to opening the scoring with their first real attack in the 15th minute when Steve Robinson released Boyd down the right and the £500,000 signing produced a beautiful centre across the goalmouth that just evaded the onrushing Rowan Vine.
Three minutes later Town felt they should have had a penalty as Boyd flicked on Leon Barnett's long ball for Brkovic to nip in behind Damion Stewart only to tumble under the defender's challenge - but referee Andy D'Urso ignored the home side's spot-kick claims.
Lewis Emanuel then slammed a 20-yard drive wide in the 19th minute after Vine had cut back inside from Robinson's ball down the left.
Luton kept attacking and a minute later they had another strong penalty appeal turned down when Emanuel drilled in a low cross-shot from the left and Carlos Edwards fell under close attention from Marcus Bignot only for match official D'Urso to again dismiss the protests from the Luton players.
Both sides continued to stretch every sinew in a blood and thunder encounter until the visitors took the lead after a questionable free-kick was awarded on the edge of the Luton penalty area in the 33rd minute.
Gareth Ainsworth appeared to take a delayed tumble after Brkovic's challenge and, from the resulting Cook free-kick, Smith ghosted in to touch home his fifth goal in seven starts for the R's from six yards.
The goal spurred on QPR and Kevin Foley had to be alert to deny Smith a second with a smart block from the 19-year-old's 15-yard shot in the 38th minute after good work from Ainsworth on the right.
A minute later Blackstock found a yard of space in between Foley and Heikkinen from Bailey's pass, but fired over from 15 yards.
It was the Hatters that found the net next, however, as Rangers dallied in first-half stoppage time allowing Robinson to seize on to Foley's quick throw down the right touchline and square for Boyd to drill home his first Luton league goal at the near post, eight yards out.
Incredibly Town found time to hit another with the last touch of the half as David Bell was given space to produce a perfect curling cross from the left for Brkovic to leap and glance home a header into the bottom corner from 12 yards.
Luton might have gone into the dressing room in buoyant mood at the break, but that all changed in the space of two minutes shortly after the start of the second period.
The visitors drew level in bizarre circumstances in the 51st minute when Brkovic and Heikkinen both got in each others' way from Cook's corner and the ball bounced off the head of Town's Finland international and into the bottom corner of his own net.
Two minutes later another corner again proved Luton's undoing as Smith picked out Nygaard on the edge of the area to send a diving header into the goalmouth where Blackstock swept home completely unopposed.
Town looked shell-shocked and, after a period of nervousness, almost got back on level terms in the 65th minute when Brkovic swivelled and crossed from the left for Edwards running in at the far side of the penalty area, but the Trinidad & Tobago man could only volley into the turf and up and over the crossbar.
As the game became stretched with Town searching for a leveller, Rangers created openings and Blackstock prodded Cook's cross past the post in the 75th minute before Smith curled just past the upright with a 20-yard free-kick four minutes later after himself being upended by Robinson.
Luton huffed and puffed, but couldn't find a way through as QPR saw out the win.