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The visitors came closest to scoring in the opening exchanges, but Alex Oxlade-Chamberlain could only put his effort wide from the edge of the box.
However, it was Everton who broke the deadlock in the 19th minute when Coleman arrived unmarked at the back post to head Gareth Barry's cross into the roof of the net.
They should have doubled their lead moments later when Kevin Mirallas was played through on goal by Naismith, before dragging his shot past the post.
On the half-hour mark, Oxlade-Chamberlain had another opportunity for the Gunners, but again he failed to find the target.
Five minutes later, it was Everton who threated the goal once more as Mirallas curled a free kick from range just past the top corner.
In the final minute of the half, Everton scored their second when Naismith was slipped into the box by Romelu Lukaku before slotting through the legs of Wojciech Szczesny, but replays showed that the striker was offside.