Arsenal pressured the Belarusian club - who hadn't played a competitive match since December 13 - early in the match but BATE weathered the storm and began to take advantage of the Gunners' lapse defending.
Lacazette's dismissal came after he aimed an elbow at Aleksandar Filipovic after the Serbian defender had fouled the France forward, who will miss the second leg.
"Letting the team down like that is the worst feeling", he wrote.
Following the match, Arsenal manager Unai Emery said he wasn't anxious about the result and trusted that his players would be able to turn the tie around in the second leg. "There are two matches".
"Our target is clear: to play next year in the Champions League", said Emery.
"You try, you play and sometimes you go home with the result you don't want, but you have to pick yourself up and make sure that you're ready for the second leg".
"I trust in our work to prepare for the next match and to have a different result at Emirates Stadium".
The Premier League side must now overturn the 1-0 deficit back at Emirates Stadium next week. We know it's a very big challenge, this last 32 against BATE and then we have to win other matches.
"Today was one of those games where if you play another couple of hours we wouldn't have scored, but there are games like that in football", he added.
"We got the result we didn't want".
BATE will travel to London for the second leg at the Emirates Stadium on February 21.