Moments later, Aleksandr Hleb tested the Kiryat Shmona goalkeeper from the edge of the penalty area.
BATE continued to press on for the opening goal of the encounter, with Dmitri Mozolevski coming close twice in two minutes.
The hosts eventually broke the deadlock in the 29th minute when Vitali Rodionov found the target after being released by Mozolevski.
The Belarusian champions continued to dominate proceedings in the second half, as Matovic was sent off in the 74th minute for a second bookable offence.
BATE duly took advantage of the numerical superiority four minutes later, as Volodko passed to Rodionov, who produced a smart curling finish for his second goal of the game.
The second leg of this tie is in Israel next week.