The Premier League side were undone by a strike from former Chelsea man Bertrand Traore, who netted the winner in the reverse fixture at Goodison Park two weeks ago, and further goals from substitute Houssem Aouar and one-time Toffees target Memphis Depay.
Caretaker boss David Unsworth admitted this week he would 'probably' need to beat Lyon and then Watford in Sunday afternoon's (Monday morning AEDT) league game in order to have a chance of securing the job permanently, but his decision to field a makeshift line-up largely comprising youngsters and back-up players did not suggest confidence in his side's chances.
Lyon were the better team throughout and Everton would undoubtedly have been shaken towards the end of the first half when Cuco Martina was taken from the field on a stretcher after landing heavily on his back.
The visitors' resistance lasted until the 68th minute, when Traore capitalised on a loose ball in ruthless fashion, before Aouar's composed strike and Depay's header made the win safe for Lyon.
The Ligue 1 side need only two points from their remaining two matches to book their place in the knockout phase, but Everton, who had Morgan Schneiderlin sent off in the closing stages, cannot progress to the next round due to Atalanta's 1-1 draw at Apollon Limassol.
Lyon controlled the opening minutes but Everton fashioned the first half-chance, with Gylfi Sigurdsson, starting as a false nine, snatching at a shot inside the area.
Jordan Pickford made the first good saves of the match in the 24th minute, keeping out a header from Traore before getting up quickly to tip away Nabil Fekir's effort on the rebound.
Depay stung Pickford's palms with a dipping free-kick before there was a lengthy stoppage in play to allow Martina to be treated, the fullback eventually being carried off in a neck brace after having fallen heavily when challenging for a header with Maxwel Cornet.
Lyon resumed their attacking intent during seven minutes of first-half injury-time, with Pickford keeping out a Depay drive from 20 yards after Marcelo had glanced a header wide, although Idrissa Gueye was denied by Anthony Lopes in the first minute of the second half when he should really have done better.
Sigurdsson flashed another effort wide but Lyon's breakthrough came at last midway through the second period. Schneiderlin challenged Tanguy NDombele and the ball broke through the static Everton defence and into the path of Traore, who took it around Pickford before slotting home.
A Sigurdsson free-kick had Lopes worried but Everton's hopes of a comeback were killed off with 14 minutes left. Mariano Diaz latched latched onto a long ball in the visitors' box and, after Pickford did well to save at his feet, Depay collected the rebound and cleverly teed up Aouar to curl a first-time finish into the bottom-right corner.
Schneiderlin earned a second yellow card for a needless lunge on Jordan Ferri before his former Manchester United team-mate Depay capped a fine outing for Lyon, the Netherlands international powering a header from Ferri's cross past Pickford, despite the keeper getting his hands to the ball.
It compounded another miserable European trip for Everton, who have won none of their four games and sit bottom of the group with two matches left to play.