Striker settled contest after Esteban Cambiasso and Jeffrey Schlupp had netted for the Foxes and a Wes Morgan own goal and a Leroy Fer strike had offered replies for the home side.
In-form forward Austin was on hand to force home the winner with a close-range header in the 73rd minute, his seventh league goal of the season ensuring Leicester is now bottom.
Austin's goal came just six minutes after Jeffrey Schlupp had pulled Leicester level, and lifted QPR above both its opponent and Burnley.
Esteban Cambiasso's second goal in Leicester colors had given Nigel Pearson's men the lead four minutes into a fast-paced first half, but QPR turned it around to go in ahead at the interval.
Leroy Fer found the net with a close-range half volley in the dying moments of the opening period, after the home side had leveled in the 37th minute through Wes Morgan's own goal.
The visiting club rallied well in the second period and equalized through Schlupp's strike from just outside the penalty area, but Harry Redknapp's outfit soon regained the advantage and duly held on for a third triumph of the season.
Leicester has endured a dire run of form in recent weeks – going seven games without a win prior to this fixture – but made an encouraging start when Cambiasso scored the team's first goal since early October.
The veteran midfielder reacted quickest to a loose ball on the edge of the box and placed a sublime finish in off Robert Green's right-hand post after Schlupp and Leonardo Ulloa had combined well down the left flank.
QPR responded well to the early blow, with both Niko Krancjar and Karl Henry going close, while returning Chilean duo Mauricio Isla and Eduardo Vargas caused havoc on the right.
Yet the home side, which has the league's worst defensive record, was always susceptible to Leicester's swift counterattacks and would have been two down if not for Steven Caulker's timely interception, which denied Riyad Mahrez.
QPR should have been level when Henry broke through the visitor's defense, but the midfielder's hesitation allowed Leicester to clear.
However, Leicester captain Morgan then put through his own net, under pressure from QPR top-scorer Austin, after Caulker had broken down the left and prodded a low cross into the six-yard box.
Leicester almost retook the lead a minute later when Jamie Vardy's powerful effort was parried away by Green, but it was Redknapp's side which went in ahead at halftime thanks to Fer's first goal for the club.
The Dutch midfielder was in the perfect position to power in a rebound after Austin's header had been well saved by Kasper Schmeichel.
Leicester responded well after the restart, and was unlucky not to equalize when Vardy's glancing header clipped off the top of the bar.
The club grabbed a deserved equalizer after 67 minutes when the versatile Schlupp flicked up the ball on the edge of the box before drilling home a superb half volley.
The sides were not level for long, though, as Austin nodded in after Leicester failed to clear a scrappy set piece.
Matty James saw an effort sail inches over the bar amid late Leicester pressure, but QPR held firm to secure a valuable three points.