Walcott put the team on the way to a win with an early double and Rob Holding added to the lead, while Mirko Ivanic got on the board for the Belarusian hosts in the first half.
Giroud scored his 100th goal for Arsenal early in the second half, but Mikhail Gordeichuk gave the hosts renewed hope after 67 minutes. Despite some spotty defending, the Gunners held on for the win.
BATE made a solid start to the match, and Ivanic fired the first shot towards goal, but after the promising start, it was one-way traffic for the visitors.
Walcott hit the post after eight minutes but made up for his miss a minute later, converting in two tries after he was set up by Jack Wilshere. Per Mattias Karen of ESPN FC, the latter could be a major weapon for the Gunners this season.
Wilshere had a goal ruled out for offside, and Ivanic perhaps should have pulled his side level―firing wide again―before Walcott struck again in the 22nd minute, courtesy of a dreadful clearance from goalkeeper Denis Shcherbitski.
And just three minutes later, Per Mertesacker nodded a ball down for Holding to make it 3-0 off a corner. Bleacher Report's Karl Matchett was not impressed at all by the hosts, as he tweeted: "BATE aren't really very good are they."
But BATE did hit back just before the half-hour mark, as Ivanic finally found the target after beating Mohamed Elneny in a duel. The goal gave the hosts confidence, but despite some late pressure in the half, Arsenal went closest to scoring through Holding, who hit the post.
Giroud converted a penalty in the 49th minute after Reiss Nelson was brought down inside the box, bagging his 100th goal for the Gunners. Per bet365, he did so faster than Robin van Persie, needing just 233 games.
Walcott wasted a chance to score a hat-trick, opting for glory instead of playing in Wilshere. A goal fell at the other end of the pitch, where Gordeichuk took advantage of some sloppy defending to make it 4-2.
The hosts took plenty of initiative late, and Aliaksandr Volodko nearly made it 4-3 with 10 minutes left to play, pulling an effort wide.
Substitute Marcus McGuane gave Walcott another golden chance at a hat-trick, but again the speedster missed. Manager Arsene Wenger gave Edward Nketiah some late playing time, but there would be no more goals.