Theo Walcott netted twice for the Gunners and Rob Holding was also on target as they eased to a second victory in as many matches in Group H.
Arsene Wenger's side beat Cologne 3-1 at home in their first outing a fortnight ago, and the Germans lost again on Thursday, going down 1-0 at home to Red Star Belgrade.
Wenger made nine changes to the Arsenal team after they travelled to eastern Europe on the back of a 2-0 win over West Bromwich Albion on Monday.
Only Shkodran Mustafi and Mohamed Elneny kept their places, while a number of youngsters featured with Joe Willock being handed a first senior start.
It was two comparative veterans who combined for Arsenal's early opener, though, with Jack Wilshere setting up Walcott to head in from close range.
Walcott made it 2-0 midway through the first half after an awful error by BATE goalkeeper Denis Scherbitski, who miscued a clearance straight to the England forward.
Holding then bundled in his first Arsenal goal but a superb header by Mirko Ivanic allowed the Belarusian champions to pull one back before the half-hour.
Holding and Walcott hit the woodwork and Giroud converted from the penalty spot early in the second half, becoming the 19th player to reach a century of goals for Arsenal.
That was not the end of the scoring, though, as Mikhail Gordeichuk got the hosts' second of the night.
"When you score four goals away from home, you're always likely to win," Walcott told BT Sport. "There was a chance for me to take the penalty for a hat-trick, but knowing Olivier was on 99 goals, it was a great opportunity for him so that's why he took the penalty."
Spaniard Hector Yuste snatched Cypriots Apollon a point at Everton.
Everton were stunned less than 12 minutes in as Adrian Sardinero took advantage of a mistake from Ashley Williams to bundle into the net.
But just as the crowd were starting to get on the home side's backs, Wayne Rooney pounced on a woeful back-pass to score his third goal since returning to Goodison Park from Manchester United.
Winger Nikola Vlasic, 19, came off the bench and slotted into the corner midway through the second half to score his first Everton goal.
But after Valentin Roberge was red-carded, Ronald Koeman's out-of-form hosts switched off and Yuste flicked in a header from a free-kick.
Everton are now bottom of Group E, three points behind leaders Atalanta, who drew 1-1 against Lyon.
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