After an extremely quiet opening period, the Chile forward grew into the game and caused the Seagulls all sorts of second-half problems.
The result means that the Merseyside club go into the global break in seventh place, seven points behind both Manchester City and United.
Arsenal, without a league title since 2004, extended its unbeaten run in all competitions to seven matches to move up to fifth in the standings.
But they struggled to break down a well organised Brighton team who were happy to play with everyone behind the ball and soak up the pressure.
Brighton's Solly March also hit the post in the first-half after a short free kick.
The France worldwide became the first Arsenal player to score in his first three home games for the club in the Premier league era.
1130: Jack Wilshere's performance in Belarus isn't enough to earn him a Premier League start as he drops down to the bench along with Theo Walcott and Olivier Giroud, with Alexis Sanchez and Alexandre Lacazette very much leading the way for Arsenal. While one of those goals was from the penalty spot, he has started very well and has scored in every home match for the Gunners so far this season, including both goals on Monday against West Bromwich Albion. "But once it settles in and we go through the game again, I think there are more positives to come from the game than negatives". It was a hard week because the game at BATE was a hard game.
"We scored a wonderful goal and created wonderful chances", Klopp told the BBC. I feel as a football side we have been getting better as well.
Chris Hughton's side lie 14th after seven games.
Lacazette will be now hoping to continue his goalscoring form at home against Brighton on Sunday, to help Arsenal get a valuable three points before the global break.