Arsene Wenger's men scored three times in the last 22 minutes to move into the final four a competition they last won in 2005.
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The under-fire Mesut Ozil answered some of his critics early on when he put the home side ahead with his first goal of the year.
His last came in December’s 1-1 Premier League draw with Everton, but the Germany international showed no signs of rustiness as he finished with aplomb inside the area.
Despite the hosts enjoying much of the play from then on, Everton levelled just after the half-hour mark following a rapid breakaway.
Romelu Lukaku profited after good work from Ross Barkley and Kevin Mirallas - prodding home from close range to continue his goalscoring form since returning from injury.
Mikel Arteta restored the lead from the penalty spot, at the second attempt after 68 minutes, following Gareth Barry's clumsy trip on Alex Oxlade-Chamberlain before Olivier Giroud struck twice late on to kill the tie off.
It ended Roberto Martinez's hopes of becoming the first manager to win back-to-back FA Cups with different clubs as Everton's winless run at Arsenal in all competitions was extended to 20 matches.
Amid frantic opening exchanges, Ozil opened the scoring after just six minutes - keeping his cool to sidefoot Santi Cazorla's slipped pass into the far corner of Joel's net.
That knocked the wind out of Everton's sails somewhat, but the visitors almost equalised shortly after when Bacary Sagna blocked off Steven Piennar inside the area.
A nervous-looking Joel then tipped Sagna's centre onto his crossbar as the Spaniard struggled to settle after a lively opening.
Going forward, Everton looked dangerous to a point but, more often than not, failed to find the killer final ball on the break.
However, they eventually got their counter-attack strategy spot on as the lively Barkley scampered down the right and crossed for Mirallas at the far post.
The Belgian cut the ball back for countryman Lukaku, who had time to control before tapping home.
Barkley had the first chance of the second half, curling an effort over the crossbar after Thomas Vermaelen gave possession away cheaply to Lukaku.
Cazorla then forced Joel into a smart save to his left before Oxlade-Chamberlain went down under the challenge of Barry soon after the hour mark following some tricky footwork from the 20-year-old.
Former Everton man Arteta then put Arsenal back in front from the spot after seeing his first effort cancelled out due to encroachment from substitute Giroud.
It was Giroud himself who got on the scoresheet seven minutes from time to settle any nerves before he finished off a quick break to move Arsenal within one game of the final.