The Toffees fielded a much changed line-up, sprinkled with youth and players unable to get much of a first-team look in this season so far.
Boss Sam Allardyce stayed at home, with Craig Shakespeare in charge, and he will have been impressed with his team, none more so than two-goal hero Ademola Lookman.
Everton started brightest and Lookman really should have had them in front early on, only to be found guilty of taking too much time in the box to see his shot blocked down.
As it was, Lookman did get the breakthrough midway through the half. Kevin Mirallas and Nikola Vlasic exchanged passes on the right and a deep cross found Lookman unmarked at the back post. He did well to steer his header past the keeper and under the crossbar.
Seven minutes later and he doubled his tally in style, unleashing a fine drive from 25-yards that the Apollon goalkeeper couldn’t keep out despite getting a hand to it.
Everton were in control for most of the second half, though they didn’t guarantee the victory until three minutes from time.
Vlasic nicked the ball in behind the covering defence, showed good pace to chase it down and beat the goalkeeper with a low shot from 15-yards to complete a satisfactory evening for the Toffees at the end of a poor Europa League campaign.
Burnley are believed to be leading the chase to sign West Brom defender Craig Dawson in the summer transfer window.
Stoke could face a battle to keep Xherdan Shaqiri as reports have claimed Mark Hughes wants him at Southampton next season.
Bayern Munich have entered the race to Emre Can, who continues to talk up a move away from Liverpool this summer.