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.
"I was hoping to score soon because I came back late from the World Cup and I was lacking maybe a bit of efficiency and luck but you always need to keep working hard and keep the faith and that's what happened tonight," France striker Giroud told BT Sport.
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.
Gerrard finally beaten in Europe
Steven Gerrard's Rangers saw their 11-game unbeaten run in this year's Europa League come to a dramatic end as they threw away the lead three times in a 4-3 defeat at Spartak Moscow.
Both sides were the beneficiaries of comical own goals in a crazy first half which ended with Glenn Middleton putting the Scots 3-2 ahead, but Luiz Adriano and Sofiane Hanni both scored in the space of 71 seconds to snatch Spartak victory.
Rangers are now level on points with Spartak in an incredibly tight Group G, one point behind leaders Villarreal after the Spaniards' goalless draw with Rapid Vienna in Austria.
Elsewhere, Eintracht Frankfurt qualified for the last 32 with their fourth win from as many group games by beating Apollon Limassol 3-2 in Cyprus, while Lazio joined them in progressing from Group H as goals from Marco Parolo and Joaquin Correa secured a 2-1 win over Marseille to eliminate last season's runners-up.
Record five-time winners Sevilla needed a late penalty from Ever Banega to claim a 3-2 victory at Akhisar Belediyespor, after having a man sent off, losing a two-goal lead and seeing the Turks miss a spot-kick of their own.
Pablo Machin's side are level with Krasnodar, who beat Standard Liege 2-1, at the top of Group J.