Marcus McGuane's deflected shot gave the Gunners a half-time lead, but a double substitution at the break paid dividends for the Toffees.
One replacement, Bassala Sambou, set up the other, Anton Donkor, to equalise before team-mate Gethin Jones was carried off on a stretcher with a suspected broken leg.
Everton took the lead when Arsenal's Joseph Olowu headed into his own net, before McGuane equalised with a penalty.
But late goals from Antony Evans and David Henen (pictured above), which sandwiched the sending-off of Arsenal's Charlie Gilmour, sealed the hosts' fourth straight league victory.
Everton, five points behind leaders Liverpool, next visit Manchester City on Friday 13 October. Arsenal host Sunderland three days later.
See: Everton report