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Villa struggled to match Arsenal's tempo in the early stages and were punished as the hosts took a deserved lead in the sixth minute.
Jack Wilshere started the move with a clever flick before Tomas Rosicky picked out Alex Oxlade-Chamberlain, who delivered an inch-perfect cross to give Olivier Giroud the simple task of tapping home from close range.
The French striker could have extended his side's lead just 10 minutes later but he failed to hit the target with a header after finding some space inside the box.
Belgian striker Christian Benteke stepped up from 12 yards and saw his penalty saved by the Polish goalkeeper, but a good reaction allowed the forward to head home the rebound.
Chances were few and far between later on in the first half but Ashley Westwood came close to giving his side a surprise lead when his chipped effort was just kept out by the back-tracking Szczesny.
Santi Cazorla was introduced to the action at half time and despite some good spells of possession, Arsenal struggled to slice open the Villa defence as Paul Lambert's men continued to frustrate the hosts.
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The Emirates Stadium voiced their disapproval of the officiating just after the hour mark when a second penalty was awarded to Villa, despite Laurent Koscielny seeming to get a touch on the ball when he challenged Agbonlahor.
Benteke was once again trusted with the penalty-taking duties and he was much more convincing from the spot this time as he drilled the ball into the bottom corner.
Koscielny's afternoon got even worse just moments later when he received his second yellow card for a poor challenge on Andreas Weimann after being booked for the tackle that gave away the penalty.
Cazorla came closest to equalising for the hosts when he powered a shot into the ground but Brad Guzan responded brilliantly to tip the ball against the crossbar.
Villa put the result beyond doubt with five minutes remaining when Weimann released Antonio Luna, who raced away from the Arsenal defence before calmly slotting the ball home to secure a memorable victory for the away side.