The Gunners capitalised on their numerical advantage to snatch a half-time lead when Mertesacker headed in from close range on 43 minutes.
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Despite Olivier Giroud's late dismissal, an under-par Arsenal hung on to claim the precious three points that see them climb above Chelsea into third in the Premier League table.
Theo Walcott found the net as early as the second minute, but his effort was rightly ruled out for offside.
Ten minutes later, Sidwell was shown his marching orders, but it was the hosts who managed to create the first notable opening when Dimitar Berbatov collected an Urby Emanuelson pass and tested Wojciech Sczcesny at his near post.
Giroud came close to breaking the deadlock just after the half hour when his shot on the turn struck the base of the post, but neither the woodwork nor Mark Schwarzer on his 500th Premier League start could deny Arsenal just prior to the interval.
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Walcott's inviting free kick was headed across goal by Laurent Koscielny, and the Frenchman's centre-back partner Mertesacker swooped to nod home from two yards.
Surprisingly, though, Arsenal did not dominate after the resumption of play and instead Sczcesny was called upon again to parry away a Bryan Ruiz strike.
The Gunners are now five points clear of fifth-place Tottenham Hotspur, who have two games in hand, in the race for Champions League qualification.