Mesut Ozil gave Arsenal an early lead with his first goal in three months before Romelu Lukaku drew Everton level at the break.
Mikel Arteta restored Arsenal's lead from the penalty spot midway through the second half before substitute Olivier Giroud scored two goals in two minutes to send Arsene Wenger's men to Wembley and end Roberto Martinez's hold on the FA Cup.
Arsenal broke the deadlock after just seven minutes with their first attack of the game as Santi Cazorla capitalised on a slip from James McCarthy to find Ozil inside the box with a perfect slide-rule pass and the German coolly slotted the ball past Joel for his first goal since scoring against Everton in December.
The goal lifted Arsenal and Yaya Sanogo and Alex Oxlade-Chamberlain both saw shots from the edge of the area pushed away by Joel as the home side went in search of a second goal.
Everton went close to scoring on 10 minutes when the ball fell to Steven Pienaar inside the box, but Bacary Sagna got back to make a vital block and snuff out the danger.
Both sides were pouring forward at every opportunity and Kieran Gibbs had a sight of goal on 17 minutes when Ozil teed him up inside the box, but John Stones got back to block his shot.
Joel almost gifted Arsenal a second goal on 24 minutes when he made a mess of a cross from the right from Sagna, tipping it onto his own crossbar under no pressure at all.
Everton drew level on 32 minutes with Lukaku finding the net with Ross Barkley the provider after a stunning run and cross.
Barkley raced clear from the halfway line before picking out Kevin Mirallas with an inch-perfect pass, only for the Belgian to scuff his shot with the goal at his mercy and he was relieved to see Lukaku on hand to tap the ball home from close range.
Arsenal tried to respond and Oxlade-Chamberlain forced Joel into a good save with a powerful left-footed shot from the edge of the box which the Everton keeper did well to turn round the post.
Play switched straight up the other end and Mirallas should have done better when he found space inside the box, but his shot lacked the power to trouble Lukasz Fabianski.
Everton had a great chance to grab the lead two minutes before the break when Oxlade-Chamberlain gave the ball away to Lukaku inside his own half to leave Arsenal outnumbered at the back, but the big striker took too long and the young midfielder got back to make a vital last-ditch tackle.
Everton had another opportunity on 54 minutes when Lukaku robbed Thomas Vermaelen down the right and he picked out Barkley on the edge of the box but he drove his shot over the top.
Arsenal won a penalty on 66 minutes when Gareth Barry tripped Oxlade-Chamberlain inside the box and Arteta, after being forced to retake his spot-kick following an infringement by substutute Giroud, kept his cool to fire into the top corner past Joel.
The impressive Cazorla went close for Arsenal three minutes later when he cut in from the left and fired in a low shot which Joel did well to turn round the post.
The home side made it 3-1 seven minutes from time when Sagna crossed for Giroud to volley home from inside the six-yard box.
And the France international put the game beyond Everton with Arsenal's fourth goal two minutes later when he collected Ozil's pass and slotted the ball home past Joel.