Subbed at half-time of his only Premier League start this season, the 20-year-old scored twice inside the opening half hour to end his 20-match scoreless run in all competitions and give the Toffees their first win at the final attempt in Group E.
The visitor clearly had an eye on Monday's derby against Liverpool in handing starting debuts to young trio Harry Charsley, Fraser Hornby and Morgan Feeney.
Lookman took his opportunity to impress by heading in the opener in the 21st minute, before he unleashed a long-range rocket soon after to effectively sew up third spot ahead of the Cypriot champions.
Kevin Mirallas added late polish to an accomplished three points with a calm one-on-one finish in the 87th minute, ensuring Apollon's first home defeat of the group stage.
In the absence of Sam Allardyce, who handed the reins to assistants Craig Shakespeare and Duncan Ferguson in order to attend a medical appointment, Everton will take all the encouragement it can get from a third straight competitive victory ahead of its visit to Anfield.
The young Blues were prised open inside six minutes as Fotis Papoulis found space on the left, but the winger harmlessly guided his finish over the bar.
Lookman had an even better chance to break the deadlock when he reeled in Davy Klaassen's through pass and stepped inside Joao Pedro, only to dally over his finish and invite the smother from Tasos Kissas.
Charsley's curler next drew an impressive fingertip save from Kissas in a lively start to proceedings.
And the visitor was rewarded for their bright opening as Nikola Vlasic collected a one-two with Kevin Mirallas and dinked a cross to the far post for Lookman to head high into the net.
At the other end, Joel Robles was required to preserve the two-goal cushion with a quickfire trio of saves and the lead remained intact at the break, despite Apollon's appeals for a penalty on the stroke of half-time.
Robles was called on again to parry Allan's firm effort soon after the restart, while Kissas covered his near post to deny Lookman a hat-trick.
Apollon, now winless in 11 group stage matches, shipped a third late on as Mirallas broke clean through and consigned the host to its first home defeat in Group E.