Stanislav Dragun nodded home Igor Stasevich's free-kick for the game's only goal just before half-time. Lacazette had a goal disallowed early in the second half before being sent off for elbowing Aleksandar Filipovic.
A resolute Bate held on for the win, their first over an English side at home in European competition. The Belarusian champions will now take a slender lead to Emirates Stadium for the second leg next Thursday, writes BBC sports.
Aliaksei Baga's side were playing their first competitive match since December and doggedly held on to their lead in the second half. Arsenal, in contrast, managed just three shots on target despite enjoying nearly 77% possession.
Even the second-half introduction of top scorer Pierre-Emerick Aubameyang and January signing Denis Suarez failed to spark a response from Arsenal, who suffered a fifth defeat in eight away games in all competitions.
It was no more than Bate deserved as Arsenal faded quickly after a bright start on a bumpy pitch and sub-zero conditions in Belarus. Their misery was compounded when Lacazette needlessly lashed out at Filipovic and was deservedly sent off.