A pre-arranged medical appointment meant new boss Allardyce was not in Nicosia to witness the 3-0 win over Apollon Limassol, the Toffees' first of the group stage, but he will receive a positive assessment of his fringe players and youngsters from stand-in manager Craig Shakespeare.
The most glowing review will be reserved for 20-year-old Lookman, who scored two goals in seven first-half minutes, the second of which was a blistering strike from range to cap off a 26-pass move where each Everton player touched the ball, with Nikola Vlasic then adding a third late on.
Everton's XI, which featured two debutants – Harry Charsley and Fraser Hornby – was the youngest an English club had selected for a European group-stage fixture in eight years, and Shakespeare also brought on three teenagers for their bows, including 16-year-old Anthony Gordon.
The Toffees named a younger starting line-up against BATE Borisov in 2009 when their place in the last 32 had already been assured, and while they selected a youthful team this time because they had already been eliminated, a win meant they at least avoided finishing bottom.
Lookman, who came through at Charlton and was making his 22nd Everton appearance, showed his inexperience when he spurned a great chance in the ninth minute.
Davy Klaassen played in the winger and he tricked an Apollon defender by feigning to shoot with his left only to hesitate on his right and allow Tasos Kissas to rush out and block the shot.
The visitors went ahead in the 21st minute as an unmarked Lookman dispatched a header into the roof of the net from Vlasic's cross after he had reached the byline via a one-two from Kevin Mirallas.
If that featured some good play in the build-up, it was nothing compared to the Toffees' second.
Every Everton player touched the ball in the build-up, including goalkeeper Joel Robles, who cleared out to Mirallas on the left flank before the Belgian found Lookman inside.
Afforded space and time, he pulled the trigger from distance and his 25-yard attempt was too quick for Kissas.
Robles was kept busy before the half ended, saving from Adrian Sardinero and Fotis Papoulis, who flicked home a free-kick only to be adjudged offside as the visitors entered the interval with a two-goal lead.
Lookman could have put the game to bed in the 57th minute when he jinked into the box and saw a left-footed try that would have given him a perfect hat-trick stopped by Kissas.
Mirallas and Hornby were both denied by Kissas' legs but he would be beaten again eight minutes from time by Vlasic's low finish.
The game had long petered out, though, and Shakespeare was able to bring on Nathan Broadhead, Gordon and Alex Denny for maiden senior appearances.
None of those will be involved in Sunday's Merseyside derby when Allardyce and Everton's big hitters return, but this was evidence that after a bleak few months, there is cause for optimism about the future at Goodison Park.