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 hosts
In-form forward Austin was on hand to force home the winner with a close-range header in the 73rd minute, with his seventh league goal of the season ensuring Leicester are now bottom.
Austin's goal came just six minutes after Jeffrey Schlupp had pulled Leicester level, and lifted QPR above both their opponents and Burnley.
Esteban Cambiasso's second goal in Leicester colours 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 hosts had levelled in the 37th minute through Wes Morgan's own goal.
The visitors rallied well in the second period and equalised through Schlupp's strike from just outside the penalty area, but Harry Redknapp's side soon regained the advantage and duly held on for a third triumph of the season.
Leicester have endured a dire run of form in recent weeks – going seven games without a win prior to this fixture – but they made an encouraging start when Cambiasso scored their 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 hosts, who have the league's worst defensive record, were always susceptible to Leicester's swift counter-attacks 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 visitors' defence, but the midfielder's hesitation allowed Leicester to clear.
However, Foxes 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 who went in ahead at half-time 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 were unlucky not to equalise when Vardy's glancing header clipped the top of the crossbar.
They grabbed a deserved equaliser after 67 minutes when the versatile Schlupp flicked the ball up 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.