Bilbao squandered a host of chances at their San Mames arena before finally making the breakthrough against the Belarussian champions in the 47th minute when San Jose swept the ball home from a Susaeta corner.
Susaeta made it 2-0 two minutes from time after fine work from substitute Borja Viguera.
Both teams were already eliminated from the Champions League and Bilbao needed only a draw to make sure of third spot behind group winners Porto and second-placed Shakhtar Donetsk.
BATE head home empty handed having set a record for goals conceded in the group phase of Europe's elite club competition of 24 following drubbings at Porto and home and away against Shakhtar.
After finishing fourth in La Liga last season, Bilbao have been on erratic form this term and are 10th in Spain's top flight after 14 matches.
Runners-up to Atletico Madrid in the 2012 Europa League, they finished fourth in their group in their only previous Champions League campaign in 1998-99.
The draw for the last 32 of the Europa League is on Monday.
(Writing by Iain Rogers, editing by Toby Davis)
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