And to make things worse for Moyes 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 side.
Yurevich’s 25-yard strike flicked off the heel of youngsters Shane Duffy and squirmed past keeper Carlo Nash.
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.