Aduriz admits Bilbao were not good enough
The Belarusian champions had never beaten Spanish opposition in eight previous attempts, but went some way to putting that right with a rousing first-half display.
Unmarked defender Denis Polyakov put BATE in front after 19 minutes with a close-range volley and, although Nikolay Signevich's superb volley struck the woodwork soon after, the hosts got the second they deserved after 40 minutes thanks to Aleksandr Karnitski's deflected shot.
And although Aduriz reduced the deficit in the final minute of the first half with a smart turn and finish, Bilbao showed very little after the interval to suggest they deserved anything from the game.
Bilbao have picked up just two wins from 10 games in all competitions this season and the striker admitted that BATE were the better side.
He told AS: "We didn't start well. They were better in every area, everything. It meant we couldn't play and we have to change the situation.
"We were really bad. But some things brought us to the Champions League and we must continue to rely on it.
"Some things change. We're not doing well and we need to do the things we did last year. It's time to reflect."
In only Bilbao's second Champions League campaign, Aduriz still feels the Liga side can progress.
"Even in the bad, the group is still alive. That gives us hope and we'll fight until the end," he said.