First-half strikes from Steven Naismith and Romelu Lukaku, along with an own goal from Mikel Arteta, put the hosts on course for a first appearance in Europe's premier club competition since 2005-06, when they failed to make it past the qualifying rounds.
After winning six Premier League matches in a row, Everton are now a solitary point behind fourth-placed Arsenal with a game in hand and have a great chance to secure European football in their first season under Roberto Martinez.
Lukaku, a threat to Arsenal's backline for much of the half, doubled the lead when he cut in from the right before evading the attentions of Nacho Monreal and Thomas Vermaelen.
Arsene Wenger's men must reflect on another damaging defeat, which extends their winless run in the top flight to four matches, and face an uphill task to secure a 17th successive season in the Champions League.
In a lively opening, Leon Osman went close after two minutes with a dipping effort that sailed marginally wide of Szczesny's left-hand post, while Lukas Podolski's shot veered just off target at the other end.
Aside from a moment of madness that saw Mirallas lose possession to Olivier Giroud inside his own area, the hosts looked comfortable in protecting their lead.
Mirallas then played a key role in Everton's third, setting a counter-attack in motion and releasing Naismith. The Scot was denied by Szczesny but Arteta turned the ball home under pressure from Mirallas.
Aaron Ramsey made his first appearance since December 26 as a substitute for Arsenal, having been out with a thigh injury, but the Welshman's return only proved a minor consolation for Wenger on an otherwise gloomy day, which ended with Yaya Sanogo having a goal ruled out for offside.
