The Gunners needed to beat Frank Lampard's side and hope Tottenham lost at already relegated Norwich, but with Spurs winning 5-0 at Carrow Road theirs was a forlorn task.
Mikel Arteta's side took the lead midway through the first half as Gabriel Martinelli stroked home a penalty after his shot struck Alex Iwobi's arm, with the spot-kick awarded after a video assistant referee check.
It proved to be a short reprieve as two second-half goals in three minutes ended the game as a contest. Both came from corners as Cedric Soares slotted home, then Gabriel poked in after Everton had initially cleared.
Martin Odegaard added a late fifth as Arsenal finished in fifth place, two points behind Tottenham. Though they missed out on the top four, their fans were at least able to celebrate a return to the Europa League after missing out this season.
Everton's record-equalling 13th away defeat, meanwhile, meant they finished four points clear of the relegation zone.
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