Olivier Giroud followed up his goal in Tuesday's Champions League win over Bayern Munich by heading the Gunners in front after 36 minutes of Saturday night's Emirates Stadium encounter.
Laurent Koscielny doubled the advantage shortly afterwards, but a deflected strike from Ross Barkley pulled Everton back into contention before the break.
Giroud, Romelu Lukaku and Mesut Ozil all hit the woodwork in the second half but Arsenal held out for a fourth straight win, with Gareth Barry sent off late on for the visitors.
The result continues a fantastic month for Arsenal after victories over Manchester United, Watford and Bayern and sees them move a point ahead of Manchester City, who play at rivals United on Sunday.
As the clock ran down, Everton continued to press and Deulofeu found himself in space on the right side of a crowded penalty area but Cech was out quickly to make a superb block.
