Roberto Martinez's side - already confirmed to finish first in the pool - was significantly weakened and the visitors took full advantage at Goodison Park to record their only win of the group stage.
Marat Izmailov pulled the strings in midfield and set up the game's only goal, cleverly scored by Ricardo Laborde in the 30th minute.
Everton, who named a bench exclusively of teenagers, were at their best when Arouna Kone - leading the line in his first appearance since October last year after overcoming a knee problem - had the ball.
The Ivorian was a regular threat but the hosts - who also gave starts to youngsters Luke Garbutt, Tyias Browning, Ryan Ledson and Conor McAleny - lacked cutting edge and rarely looked like scoring after the first 30 minutes.
Krasnodar's win lifted them above Lille, who lost 3-0 to German side Wolfsburg, and into third in the group.
Bryan Oviedo - still on the comeback trail from a broken leg - and goalkeeper Joel Robles made rare starts for Everton, while Gareth Barry, suspended for the club's next Premier League match, deputised at centre-back and Steven Pienaar added experience.
Everton's start was far from ideal, though, as winger Christian Atsu - on loan from Chelsea - hobbled off in the eighth minute clutching his hamstring, with England Under-17 international Kieran Dowell on to replace him.
Antolin Alcaraz almost gave the home side the lead in the 10th minute when he nodded Garbutt's corner just wide but the chance failed to awake the crowd from their near-permanent slumber during the match.
Kone stung the palms of Krasnodar goalkeeper Andrey Sinitsyn from 25 yards in the 15th minute before Krasnodar created their first meaningful chance - a powerful Wanderson effort that was well kept out by Robles.
The Russian Premier League outfit grew into the game from there and Izmailov begun to show his class.
Garbutt was Everton's brightest spark down the left and stormed to the byline before his half-cleared delivery allowed Pienaar to try his luck from distance, but again Sinitsyn was equal to the task.
And the visitors made Everton pay for their missed chances when Laborde gave them the lead.
Izmailov played him in with an excellent ball over the top and the scorer got there ahead of the onrushing Robles before turning away from goal, then spinning sharply and catching the goalkeeper unaware at the near post.
It was a smart finish but perhaps Robles should have done better, as Everton headed into the break behind.
Kone seemed desperate for a goal on his return and after an excellent turn, fired straight at Sinitsyn in the 50th minute, while Ragnar Sigurdsson made a smart block to deny Dowell after good build-up work from Oviedo.
Izmailov could have sealed Krasnodar's win in the 72nd minute, but shot inches wide to end an excellent move from the visitors, before Kone nodded a free-kick over.
But Krasnodar's 14 travelling fans - housed in corporate hospitality by Everton - went home happy as the hosts ultimately ran out of answers.
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