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Arsenal finished the half on top and twice forced Everton goalkeeper Tim Howard to make a save.
The second half started with much more action and Santi Cazorla had a good chance for Arsenal but Jack Wilshere's pullback was behind the Spaniard, who could not generate enough power with his header.
Everton had their first shot on target in the 54th minute when Steven Pienaar stung the palms of Szczesny with a fierce drive.
Moments later Kieran Gibbs's deep cross was volleyed on the stretch by Ramsey, with Howard pushing the ball round his near post.
Ross Barkley was next to have a shot from distance but he, like Pienaar, fired powerfully straight at Szczesny.
Arsenal manager Arsene Wenger made a triple substitution to shake up his side and it almost worked as one of the changes, Mathieu Flamini, drilled a yard or so wide.
Mirallas then had a near-post drive deflected into side-netting by Laurent Koscielny's sliding tackle. From the resulting corner, Sylvain Distin had a free header at the back post but he could not keep it down.
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Arsenal went ahead with 10 minutes on the clock when Tomas Rosicky crossed for fellow substitute Theo Walcott, whose header back across goal was missed by Giroud but side-footed into the roof of the net by Mesut Ozil.
The Gunners had been ahead for just a couple of minutes when Everton substitute Deulofeu silenced the majority of the Emirates.
The Barcelona loanee had only been on the field for four minutes when he found the far corner with a dipping shot from the right side of the area.
In the final moments both sides looked like they might be able to nick it and it was Giroud who came closest with a spectacular effort.
The French striker let fly on the volley from 30 yards and struck the upright with a swerving effort that Howard had no chance of stopping.
A point sees Arsenal edge five clear of Liverpool and Chelsea at the top, while Everton remain fifth, two further adrift.