Alexis returned to the scoresheet for the first time in eight games.
France striker Giroud's fifth goal in as many games set Arsenal on course, before Sanchez ended a seven-game drought.
Giroud's 13th goal of the season calmed Arsenal who had headed into the break unsettled and unnerved by a spirited effort from relegation battlers Rangers.
Austin set up a fraught finish with a fine 20-yard strike on the turn for Rangers, but Arsenal were still able to cling on for victory.
The Gunners' superior class and counter-attack quality proved decisive, with Sanchez influential in the build-up to Giroud's goal before pilfering his 19th finish of the campaign.
Arsenal were denied a clear penalty when Karl Henry pulled back Mesut Ozil to stop the German tapping in from two yards, but the visitors were able to shake off referee Kevin Friend's error.
Arsenal's 10th win in 12 games in all competitions keeps them third in the table, even though the Gunners spent the entire first half at Loftus Road struggling for any kind of control.
Rangers wasted little time squeezing Arsenal right across the pitch, with stand-in skipper Bobby Zamora offering a talismanic performance.
The veteran target man teed up strike partner Austin for two promising 20-yard efforts, but the in-form poacher could not provide a telling enough finish on either occasion.
Austin's second long-range effort had David Ospina frozen to the spot, with the Arsenal keeper forced to watch the ball whistle just past his right-hand post.
Matt Phillips ran Arsenal's right flank ragged in the first half, and his most dangerous cross almost crept past Ospina and into the net.
After Austin and Zamora both failed to reach the whipped ball, Ospina was forced to turn the cross round his left post.
Arsenal were restricted to just one tangible goal-scoring opportunity in the first-half, Santi Cazorla hitting the target but failing to test Rob Green.
The Spain midfielder will no doubt have been frustrated not to catch his effort cleanly enough to blast past the in-form QPR keeper Green.
Arsenal finally found their missing fluency after the restart, Rosicky immediately carving in behind Sandro to tee up the onrushing Mesut Ozil.
The Germany midfielder could not convert Rosicky's cross from close-range, and wound up complaining he was pulled back by Karl Henry.
Rangers' defensive midfielder certainly made contact, and was fortunate not to cost his side a spot-kick.
Sanchez weaved and stepped his way to another Arsenal chance as the early part of the half unfolded, but the Chile hitman curled his effort comfortably wide.
Austin had another 20-yard effort blocked as QPR attempted a rally, before Phillips fired a piledriver just the wrong side of the near post.
When Arsenal's lead finally came, the Gunners struck through Arsene Wenger's time-honoured tradition of the counter-attack.
Ozil fed Sanchez to saunter from halfway to the edge of QPR's area, and the home defence backed off and allowed the Chilean to set Gibbs through on the overlap.
Even if Sanchez's reverse pass was neat, it should have been obvious: instead QPR let Gibbs steam through and produce a low cross that Giroud turned home via a deflection.
Sanchez then doubled Arsenal's lead inside 10 minutes, turning Darnell Furlong inside out before dispatching an accurate low shot past Green.
Just when Arsenal expected to coast home, up popped in-form striker Austin to blast home on the spin and hand Rangers a lifeline.
The sharp poacher's 15th goal of the campaign raised home hopes at the death, but - despite the rally - Arsenal still tiptoed across the line.
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