The 20-year-old's brace and a late strike from Nikola Vlasic gave a much-changed Toffees side a solid victory in Cyprus.
Everton's win ensured that the Merseyside club, already out of the competition, would not finish their Group E campaign without at least one win on board.
As well as praising his performance, Lookman also declared himself happy with life under new manager Sam Allardyce, who was not present at the game due to a medical appointment.
"It was very important. To get three points today and a clean sheet was fantastic," Lookman told reporters. "I should have had a hat-trick, but it's one of those days - the keeper saved a couple of my chances.
"It's been good [under Allardyce]. Different training, different tactics. I'm enjoying it. Previously with the results and mood in the camp, players' confidence wasn't as high as it is now. We're getting there."
Up next for Everton is Sunday's Premier League derby with Liverpool.