An early own goal from Zakhar Volkov set the Gunners on their way, and headers from Shkodran Mustafi and Sokratis Papastathopoulos sealed the club's progress through to the next round by a 3-1 aggregate scoreline.
A cross from the right from Pierre-Emerick Aubameyang did not come close to finding a teammate, but Volkov was only able to divert the ball into his own net from eight yards out.
Arsenal remained on the front foot with Iwobi curling marginally wide of the target, but BATE almost stunned the home crowd with an equaliser as Stephan Lichtsteiner was forced to clear Stanislaw Dragun's half-volley off the line.
The tempo of the game eventually slowed down, but Arsenal stayed in control and Granit Xhaka almost doubled his side's lead with a 25-yard strike which was tipped over by Scherbitski.
Aubameyang had been relatively quiet in the final third but just before the half-hour mark, the Gabon international scuffed a volley straight at Scherbitski before directing a header onto the roof of the net.
Arsenal continued to create opportunities on a regular basis and shortly after Mesut Ozil had sent a spectacular volley marginally over the bar, the North London side netted their second.
An outswinging corner from Xhaka found Mustafi all on his own in the penalty area, and the German was able to power a header past Scherbitski from close range to put his team ahead on aggregate.
Arsenal went back onto the front foot after the restart and Matteo Guendouzi went close with a shot from distance, but BATE should have regained their advantage in the tie shortly afterwards as Maksim Skavysh fired straight at Petr Cech from 12 yards out.
That near miss ensured that Arsenal showed no complacency for the rest of the contest, and Iwobi was soon denied by Scherbitski from 15 yards out before Henrikh Mkhitaryan headed wide at the back post.
The home side were forced into a change due to an injury to Laurent Koscielny, but his replacement Papastathopoulos put the result beyond doubt with a header into an empty net after Scherbitski had misjudged a corner.
Ozil and Aubameyang both had opportunities to add to Arsenal's lead, but the German sliced a volley wide of the near post and his teammate dinked a one-on-one effort tamely off target.
Aubameyang also saw a low free kick well held by Scherbitski, but the closing quarter-of-an-hour was played at a leisurely pace as Arsenal remained on course for another lengthy run in this competition.
ARSENAL (4-2-3-1): Cech; Lichtsteiner, Mustafi, Koscielny (Papastathopoulos 56'), Monreal; Guendouzi (Torreira 64'), Xhaka; Mkhitaryan (Suarez 78'), Ozil, Iwobi; Aubameyang
BATE BORISOV (4-3-3): Scherbitski; Ryas, Volkov, Filipenko, Filipovic; Simovic, Dragun (Berezkin 64'), Baga (Hleb 58'); Stasevich, Milic, Skavysh (Dubajic 72')