In the other Group L match, Vidi FC also managed a 1-0 victory over PAOK to rank second on the log with six points - six behind table toppers Chelsea.
Giroud had not scored for Chelsea since May, as the striker went 794 minutes without finding the back of the net for the Blues.
Olivier Giroud's second-half victor away to BATE did little to brighten Maurizio Sarri's mood following a sub-par display from Chelsea. Ten-man Sevilla won 3-2 at Akhisar thanks to a late penalty, but Anderlecht was eliminated.
And Sky Sports pundit Merson says that showed exactly why Chelsea need to sign a new striker in January.
The 39-year-old has been praised by manager Galtier who expressed delight that the young ones would be impacted by the vast knowledge of the game by the former striker. "They were unlucky and could have drawn the match".
While the former Juventus striker netted seven times in seven games to make an electric start to his Blues career, the Spain worldwide managed just three times in their last 24 appearances in all competitions.
Sarri is now hungry for the additional point needed to secure top spot in the group, to postpone thoughts of Europe until the last-32 stage.
"I think we didn't play very well, I didn't like the first half because we moved the ball only horizontally and slowly, so it was hard to be unsafe".
The Brazilian was due to take part in at least some of Saturday's training session and it appears he has come through that as he makes the party to travel south.
In Group G, Spartak Moscow beat Rangers 4-3 in a thriller after the Scottish club had led 3-2. "It was a very hard day for me".
But as Chelsea grew more assured, the hosts retreated, and early in the second half Emerson's cross allowed Giroud to nod the ball in at the near post, his first goal since May.
"He's going through one-on-one with the keeper to get his hat-trick".