Stanislaw Dragun's header shortly before the half-time whistle proved to be the difference as Arsenal toiled on a poor pitch and in freezing temperatures in Belarus.
The North London outfit also had Alexandre Lacazette dismissed for an elbow during the closing stages, and Unai Emery's side must come from behind in next week's second leg in order to progress through to the last 16.
Arsenal wasted little time in going onto the front foot and Henrikh Mkhitaryan should have put the visitors in front after two minutes, but the Armenian was superbly denied by Denis Scherbitski after getting on the end of Sead Kolasinac's low cross.
Mkhitaryan then turned provider for Nacho Monreal to glance a header wide of the far post, and it appeared only a matter of time before the Gunners would take the lead.
Arsenal went up the other end with Lacazette heading wastefully over from 10 yards out, while BATE's Nemanja Milic failed to hit the target from close range after good work from Skavysh.
The Premier League side rallied before the break and appeared to be the more likely to open the scoring, but BATE went ahead shortly before the half-time whistle as Dragun sent a header into the far corner from 12 yards after meeting an outswinging free kick.
Arsenal made a bright start to the second half and the visitors managed to find the back of the net in the 56th minute when Lacazette converted Kolasinac's cross from close range, but the forward was adjudged to be marginally offside.
BATE remained content to play on the break and Ihar Stasevich forced Cech into a save to his right with a volley from 25 yards, and it was an opportunity which contributed to convincing Emery that he needed to make changes.
The night went from bad to worse for Arsenal as they were reduced to 10 men after Lacazette swung out an elbow when attempting to break free of Aleksandar Filipovic on the flank.
That proved to be the final incident of the contest and although Arsenal will remain as favourites to make the next round, they will be forced to name a strong side rather than taking the opportunity to rest some of their key men.
BATE BORISOV (4-3-3): Scherbitski; Ryas, Volkov, Filipenko, Filipovic; Baga, Dragun, Hleb )Berezkin 58'); Milic (Dubajic 69'), Skavysh (Moukam 80'), Stasevich
ARSENAL (4-3-3): Cech; Mustafi, Koscielny, Monreal; Maitland-Niles (Aubameyang 68'), Guendouzi, Xhaka (Torreira 69'), Kolasinac (Suarez 74'); Mkhitaryan, Lacazette, Iwobi