Already assured of qualification and unable to improve their group position, David Moyes named an extremely youthful team that showed nine changes from their previous Premier League engagement - the manager clearly eyeing the upcoming league clash with Birmingham.
No less than five players were making full debuts with 16-year-old Jake Bidwell becoming the youngest Everton player to play in Europe.
And they held their own until the 75 minute mark when Aleksandr Yurevich drilled in a low shot from 25 yards that deflected past Carlo Nash.
An equaliser wasn't forthcoming but the young Toffees' performance against the Belarusian champions - and Leon Osman's 75-minute return from injury - were plusses.
Soon after kick off though, just seven minutes in fact, the inexperienced line-up became even more so as Jack Rodwell was replaced by another debutant - Hope Akpan - after picking up an apparent hamstring strain.
After 15 minutes BATE's Vitali Rodionov was wayward from long distance with the first real effort at goal. Osman replied with much better skidding effort but Sergei Veremko gathered.
Halfway through Sergei Sosnovski's header was blocked away from goal by his own teammate. Then Igor Shitov blasted miles over from a corner and Nash saved Yurevich's firm effort.
Osman played in Kieran Agard just after the half hour and the young forward fired marginally over the bar.
And just before half time it looked like Sergei Krivets would open the scoring but Shane Duffy was on hand to deflect his effort away for a corner.
The Northern Irishman produced an even better block at the start of the second half, again denying the Belarusian playmaker.
Play continued to be disjointed and Yakubu's wayward effort was typical of the game as a whole.
Next the impressive Duffy cleared off the line and but soon BATE did take the lead.
Yurevich drilled a low left foot effort that flicked Duffy and beat Nash from 25 yards, with 15 minutes left to play.
Moyes immediately withdrew Hibbert from the action and replaced him with another teenage debutant, Shkodran Mustafi.
Baxter's free kick was a yard wide and Agard's low shot from the edge of the box even closer but a leveller wouldn't materialise.
