Olivier Giroud's first goal in 794 minutes of club football was one of few positives from a lethargic display that could well have cost the Premier League side.
BATE were denied by the woodwork three times, including once in stoppage time as the hosts went within inches of rescuing a result in frosty conditions.
Chelsea ultimately clung on to move within a point of confirming their status as Group L winners, although their coach was left concerned.
"We didn't play very well. I didn't like the first half because we moved the ball only slowly," Sarri told reporters.
"It was very difficult to be dangerous, but we were better in the first 25 minutes of the second half. We didn't manage the last 15 minutes well.
"The best thing is the result. It wasn't very easy to play a good game here, with the conditions."
Giroud's well-judged header from Emerson Palmieri's cross in the 52nd minute did provide Sarri with the rarity of two strikers in scoring form following Alvaro Morata's brace in the weekend win over Crystal Palace.
The goal opened Giroud's account under the Italian and was his first for Chelsea since May.
"I am pleased for him. He is very important for us," the Blues boss said.
"I know very well a striker wants to score. For his confidence it is very important."
France striker Giroud insisted he lost no belief in his ability during the drought.
"I maybe lacked a bit of efficiency and a bit of luck sometimes, but you always need to keep working hard in training and keep the faith," the 32-year-old told BT Sport. "That's what I did tonight."