Denis Plyakov gave the home side the lead on 19 minutes when he fired into the roof of the net after a quick corner had been flicked into the defender's path.
© Getty Images
The Belarus outfit nearly doubled their lead on the half-hour mark but Mikalay Signevich smashed a volley against the post.
BATE doubled their advantage towards the end of the first period when Alyaksandr Karnitsky fired in an effort from the edge of the box to wrongfoot the goalkeeper.
However, Aritz Aduriz struck on the cusp of half time to reduce the deficit for the La Liga outfit.
Signevich could have extended the advantage to two once more on 70 minutes, but the ball somehow got stuck beneath his feet as he went through on goal.
Athletic went in search of the equaliser towards the final third of the game and Susaeta went close with a long-range effort that was tipped over the bar by BATE keeper Syarhey Chernik.