Promising midfielder Jack Rodwell was forced off after just eight minutes when he appeared to catch his studs in the turf.
It was an early blow for an Everton side, who had already qualified for the last 32.
However they stayed on level terms until the 75th minute when a goal from Aleksandr Yurevich won the game for an experienced BATE Borisov side.
It was a night when Tony Hibbert and Jake Bidwell achieved European club records for Everton.
Hibbert set a top mark of 20 appearances while Bidwell at 16 years and 271 days became their youngest player at this level.
Bidwell was one of five players making full debuts with reserve goalkeeper Carlo Nash, Shane Duffy, Kieran Agard and Adam Forshaw also included.
Not surprisingly given their changes Everton were feeling their way into the game but Agard showed good skill in the 14th minute.
However he was bundled over by Sergei Sosnovski, which earned the BATE player a yellow card.
Everton's first shot of note came in the 20th minute when Leon Osman, back after a foot injury, tested goalkeeper Sergei Veremko from distance.
BATE were beginning to knock the ball around and Nash did well to block a shot from Aleksandr Volodko in the 29th minute.
Three minutes later Everton hit back and Agard made space for himself inside the area only to send his effort over the bar.
It was becoming an open game and Maksim Skavysh tried his luck in the 36th minute but failed to trouble Nash.
Then Sergei Krivets had a clear opening for the visitors four minutes later in front of the posts.
However he failed to get any power on his effort and Nash was able to make a comfortable save.
Duffy then showed good awareness to deflect to safety a shot from Krivets a minute before the break
Everton started the second half with purpose and Ayegbeni Yakubu tried to release Agard in the 50th minute but put too much weight on the pass.
Yakubu had been struggling to get into the game but shrugged off a challenge with disdain in the 60th minute.
However his effort from distance failed to trouble the goalkeeper as it went well wide of the target.
BATE almost went ahead in the 62nd minute after Nash suffered a rush of blood to the head.
He came racing out of his area to try and close down Pavel Nekhaychik but was left stranded.
Luckily for Nash, Duff was on hand to clear Nekhaychik's shot and keep the match on level terms.
Everton were left reeling in the 75th minute when BATE made the breakthrough.
There seemed little on when Yurevich picked the ball up 25 yards out but Nash seemed unsighted as it trundled past him and into the net.
Everton hit back and Jose Baxter delivered a neat free-kick four minutes later that went narrowly wide of the post.
Then Agard scuffed a shot wide in spirited response but they were unable to get the equaliser.