Theo Walcott netted twice inside the opening 22 minutes, while further goals from Rob Holding and Olivier Giroud - his 100th in an Arsenal shirt - put the Gunners into what turned out to be an unassailable lead.
Both teams began quietly in Belarus, which was welcoming Arsenal for the first time, but Arsenal were soon able to take control of proceedings, with Walcott striking the post after latching onto Jack Wilshere's through-ball.
However, a minute later, the same duo combined again to contribute the opening goal, with Wilshere's cross being finished off by Walcott from close range at the second attempt.
Arsenal reverted back to controlling the game on the halfway line, while BATE were being reduced to pot shots from distance, so it came as no surprise when the visitors doubled their advantage.
Goalkeeper Denis Scherbitskiy sent a pass straight to the feet of the grateful Walcott, who bided his time before finding the bottom corner with his low shot.
Poor defending also led to Arsenal's third of the evening, with Holding diverting the ball into the net from close range after Per Mertesacker had towered over the BATE defence to meet Walcott's corner.
However, the hosts soon got themselves back into the match when Ivanic headed the ball into the top corner after being found by Alyaksey Ryas's cross from the right flank.
BATE remained in the ascendancy as Arsenal came under pressure for the first time in the contest but towards the break, the North Londoners squandered two opportunities to extend their lead.
Firstly, Walcott was denied a hat-trick by a stunning stop from Scherbitskiy from just eight yards out, before Holding sliced a volley against the post from inside the six-yard box.
However, three minutes into the second half, Arsenal netted their fourth thanks to a penalty from Giroud, who sent the ball into the bottom corner after Stanislaw Drahun was harshly adjudged to have fouled Shkodran Mustafi.
Despite their three-goal lead, Arsenal soon became second best for the remainder of the match and they should have been pegged back just before the hour mark, with Ospina making a brilliant save from Mikalay Signevich's header.
Ryas also went close with a fizzing strike from 30 yards out, but the home side finally had their second when some lax marking in the final third allowed Gordeichuk to rifle the ball into the roof of the net from three yards out.
With 19 minutes left, BATE should have had a third when Alyaksandr Valadzko sent a first-time effort wide of the post from by the penalty spot, with no Arsenal player offering any pressure.
While looking shaky at the back, Arsenal continued to push for a fifth goal with Giroud twice squandering efforts on his favoured left foot, but the English team were able to see out the remaining minutes with minimal fuss to remain at the top of Group H by two points.