The visiting side picked off its Ligue 1 opponent with a fine counterattacking display before the break, and managed to withstand heavy pressure to take all three points.
The visiting side went in 3-0 up at halftime following an extremely clinical first-half showing that was branded a "nightmare" by home coach Rudi Garcia.
It was hard to argue with that assessment, given that BATE had scored with three of its four shots on goal, as Aleksandr Volodko, Vitali Rodionov and Edgar Olekhnovich all found the net impressively.
Aurelien Chedjou hit back after the break, despite the fact that his header looked to have bounced on the line rather than over it, but Viktor Goncharenko's side stood firm in the face of fierce pressure from les Dogues.
It was the home side which started relatively brightly, as Salomon Kalou stung the palms of Andrey Gorbunov after four minutes with a half volley at the back post, but it was the underdog that went ahead two minutes later.
Volodko silenced the crowd with a superb strike from 30 yards that flew into the top corner past a despairing Mickael Landreau.
Newcastle United summer transfer target Mathieu Debuchy went close with a low drive six minutes later, but Gorbunov got down well to turn it away for a corner.
As the home side set up camp on the edge of its opponent's penalty area, BATE broke up field and scored a highly contentious second after 20 minutes.
Former Arsenal man Aliaksandr Hleb carried the ball a good 60 yards before laying off a cute ball out left to Aleksandr Pavlov, who then found Rodionov with a first-time pass that allowed the striker to slide the ball home. It was a wonderfully worked goal, but the home crowd was incensed upon seeing replays that revealed Rodionov was offside from Pavlov's pass.
Marvin Martin sliced a shot wide after Gorbunov made a mess of a high free kick, and Dimitri Payet's goal-bound shot was blocked after he was smartly picked out by Debuchy following a clever header from Tulio De Melo.
But again it was the visitors who struck next. Volodko troubled the hosts with a direct run before picking out Olekhnovich, whose shot was cleared away by Landreau. But the ball fell back to Volodko, who picked out Olekhnovich once again, and this time he made no mistake and sent the ball into the bottom right corner.
Coach Garcia branded the first half a "nightmare" on his way to the dressing room, and he responded by replacing De Melo and Martin with Ryan Mendes and Nolan Roux.
Kalou headed another Debuchy cross wide five minutes into the new half, before the right back saw his deflected shot clip the roof of the net.
But the team got its goal with half an hour left when Chedjou's header was adjudged to have crossed the line after crashing down off the crossbar from Payet's corner. It looked the wrong decision, but the home side will feel it was due a bit of luck following some contentious calls in the first half.
Les Dogues had the bit between their teeth from then on, and Mendes saw a shot saved at the far post after a prolonged spell of pressure. Seconds later, Payet's swerving shot was turned away by Gorbunov as the home side really threatened to stage a comeback.
Payet forced a flying save from the keeper once more with a 77th-minute free kick, and Florent Balmont was denied by the keeper's legs two minutes later when he really should have done better when unmarked at the far post.
Hleb went close to settling it almost immediately, but his shot on the counterattack lashed against the side netting.
The home side lost momentum in the dying stages, perhaps exhausted by the frantic but fruitless 20-minute spell which followed Chedjou's reply.
It could prove to be a costly evening for the French side, which will travel to face Valencia in its next game with both sides looking for their first points.
BATE, meanwhile, will have to reproduce this performance and then some when it hosts Bundesliga giant Bayern Munich, but could pull off the mother of all shocks if the club can be as clinical as it was here.
Follow GOAL.COM USA on