The visitors took the lead after 30 minutes through Ricardo Laborde, who raced through the back line to finish past Joel Robles at an angle.
Arouna Kone did his best to bring Everton back into the game with two close efforts, but Krasnodar came nearest to getting the second goal through Marat Izmailov, who pulled wide from 12 yards.
Everton had already secured top spot in Group H on matchday five, and finish with three wins, two draws and a defeat.
Martinez gave a debut to 17-year-old midfielder Ryan Ledson, who captained England to the Under-17 European Championship title, and has led the national Under-18 side this season.
Youngsters Luke Garbutt and Conor McAleny were handed starts, while Kieran Dowell and Tyias Browning came on as substitutes to bring a youthful look to Martinez’s side.
Paraguay defender Antolin Alcaraz, making only his 11th start since joining Everton, came close to scoring his first Toffees goal early on, but his firm header from a corner brushed past the left-hand upright.
Everton were forced into an early change, with youngster Kieran Dowell coming on for Christian Atsu, who was forced off after seemingly tweaking his hamstring.
Kone, who himself has struggled with injury problems, forced a great save out of Andrei Sinitsyn in the Krasnodar goal with a flashing effort towards the bottom corner.
Wanderson gave the hosts a warning just before the opener, stinging the hands of Everton goalkeeper Robles, and Krasnodar got their goal a third of the way through the game.
Laborde silenced Goodison Park, latching on to a through ball to slot past Robles from an angle when it looked like he would cut the ball back to a team-mate.
Alcaraz saw a header sail over the bar shortly after, but following the break it was Krasnodar who were looking for the next goal.
Robles had to be sharp to block an effort low at his feet, while at the other end Kone turned sharply inside the penalty area, only for Sinitsyn to beat the ball away at his near post.
With 18 minutes remaining, Joaozinho fed Izmailov, who twisted but sent his low effort just wide of the post with Robles stranded.
Krasnodar held on for the win, but finished third in Group H, four points behind Wolfsburg.
Everton: Robles (5), Browning (6), Alcaraz (6), Barry (6), Garbutt (7), Ledson (6), Pienaar (5), Atsu (5), Oviedo (6), McAleny (6), Kone (6).
Subs: Dowell (6), Long (6), Jones (6).
Krasnodar: Sinitsyn (7), Granqvist (6), Gazinski (7), Izmailov (7), Wanderson (7), Kaleshin (6), Laborde (7), Joaozinho (6), Sigurdsson (6), Pereyra (6), Petrov (7).
Subs: Burmistrov (6).
Man of the match: Ricardo Laborde.