Giroud ended his 11-game goal drought this season by giving Chelsea a 53rd-minute lead, and the Premier League giants held on despite BATE striking the woodwork three times in the match.
The Londoners moved six points clear of Hungarians Vidi at the Group L summit and remain unbeaten in 17 matches since losing the Community Shield to Manchester City in August.
BATE started strongly and came within a whisker of taking an early lead as former Arsenal midfielder Alexander Hleb teed up Dmitri Baga for a long-range strike which crashed against Chelsea goalkeeper Kepa Arrizabalaga's post.
The visitors quickly started to monopolise possession, but despite seeing over 70 percent of the ball in the first half, they failed to even manage a shot on target against a well-drilled BATE outfit.
But Sarri's men were much improved after the break and Giroud, who also scored twice for Arsenal against BATE in the Europa League last term, broke the deadlock by meeting left-back Emerson's cross with a firm downward header.
The hosts almost levelled just five minutes later, as they hit the woodwork for second time in the match when Nikolai Signevich diverted the ball onto the crossbar.
BATE coach Alyaksey Baha, the older brother of midfielder Dmitri, was left holding his head in his hands after Aleksey Rios blazed an even better chance to equalise over the bar.
There was still time for BATE to hit the post again, as right-back Rios' brilliant dipping volley beat Kepa but bounced to safety.