After a cagey opening, the Gunners went ahead as Giroud met Mesut Ozil’s floated cross with a deft touch.
Koscielny doubled the lead two minutes later from Santi Cazorla’s ball in.
Ross Barkley’s deflected shot brought Everton back into it, but Romelu Lukaku wasted an excellent, if rare, chance to level as he headed against the bar.
Arsenal goalkeeper Petr Cech denied Gerard Deulofeu late on, but the hosts also had their chances to make the scoreline more comfortable.
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