Sarri had to endure 90 minutes of struggle as Chelsea failed to shine in Belarus.
Olivier Giroud scored the only goal as Chelsea maintained their 100% European record this season and booked their place in the knockout rounds with two games to spare.
And Sarri was not happy with the performance and revealed the result was the only element of the game he was pleased with.
Sarri said: “What pleased me about tonight was just the result. We didn’t play well.
“There were conditions to play badly – the cold, the opponents, the pitch, everything. But we could have done better.
“The first objective was to qualify. The second is to win the group. We need one more point for that.
“But it’s important because now we don’t talk again about this competition until February.
“I’m pleased for Olivier. He’s a very important player for us, and strikers want to score.”
He added: “Today I was in trouble, too. I had fever. It was a very hard day for me. I don’t know if I am improving or if it is only the temperature.”
BATE, comfortably beaten last month in London, struck the woodwork three times, including deep in stoppage time, and Eden Hazard’s services were required for longer than anticipated in Belarus.
As the game clicked into the 51st minute, Hazard was fouled and sent the resulting free-kick soaring out of play beyond the far post.
Hazard had been due to be off by then and Sarri was gesticulating wildly in his technical area, apparently towards the Belgian.
Giroud struck moments later, heading in Emerson Palmieri’s cross at the near post.
Giroud told BT Sport: “It’s true, I was hoping to score soon because I came back late from the World Cup and lacked a bit of luck. You always have to work hard and keep the faith.”
Sarri added: “I’m pleased for him. He’s a very important player for us, because he’s a very, very useful player for the team.
“But I know very well strikers want to score. And so for his confidence I think it’s very important. But, I don’t want him to be worried about scoring, because for us he’s very important, very useful.”
Chelsea would have liked to score another, but instead it was BATE pressing late on.
“They played very well,” Sarri added. “They were unlucky. I think they could draw the match.”
Chelsea have qualified with two matches to spare, the November 29 home clash with Greece’s PAOK and the December 13 trip to Vidi in Hungary.
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.
He added: “The first target in this competition was to qualify. Now we want the second target, to win our group.
“It’s very important because then we can talk again about this competition in February. In February we will have for sure another target, but the first one is very important.”