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Thirty-six minutes in, a low cross from Matt Phillips threatened to turn into a shot from the right flank, but Ospina was again able to get a vital hand on the ball as another promising move broke down.
Arsenal finished the first half strongly as Santi Cazorla stung Green's palms with a volley, and they continued to push forward after the restart.
Tomas Rosicky delivered a cross that proved just too strong for Mesut Ozil, but Alexis Sanchez then found the space to cut inside and unleash a low shot to force Green into a decent save.
QPR countered, with Phillips just inches away from breaking the deadlock on the hour mark with a half-volley which drifted just wide.
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The hosts would rue that miss because, just six minutes later, Sanchez picked out Kieran Gibbs on the wing, and even though the subsequent cross was blocked, Giroud reacted quickest to blast the ball home.
Straight after the restart, Green was tested twice again as Arsenal took complete control of proceedings, but it didn't take long for the second Gunners goal to come.
Sanchez, so instrumental in Arsenal's play so far, cut in from the left before picking out the bottom corner with a powerful finish for his first strike in seven games.
QPR, to their credit, made for a nervy finish when Austin turned into space on the edge of the box before lashing a fierce shot towards goal past the helpless Ospina.
However, it was too little too late as the Gunners saw out the game to secure all three points, which sees them remain third in the Premier League table.
Meanwhile, the struggling R's remain stranded in the relegation zone, three points from safety.