Lacazette was the only casualty from the squad that beat Chelsea 3-1 on Dec. 26, with the France international making way for Pierre-Emerick Aubameyang, but it took him just 21 seconds to score the game's only goal when he came on as a substitute.
It was Brighton that had the better chances in the first half with eight shots on goal compared to Arsenal's one. However, neither side came particularly close to breaking the deadlock.
The second half got underway with a Brighton penalty shout after Lewis Dunk was brought down in the area. However, VAR ruled that no foul was committed.
Arsenal had two quick good chances shortly after with Aubameyang being denied from close range. Minutes later Gabriel Martinelli's right-footed shot went curling just over the crossbar.
Martinelli made way for Lacazette on 66 minutes and made his presence felt immediately. Given too much space, he whipped a right-footed shot in from the centre of the box and past Robert Sanchez in goal.
"I'm proud for the boys and happy for the result. Lacazette is in a good moment, he's full of confidence," said Arsenal manager Mikel Arteta. "Today he had a sore back, he could not start the game. But he came on and won the game for us which is massively important.
"We know we've been struggling with results, [and] the fact we won against Chelsea was a big boost. To come here and get another win was massively important."
Bukayo Saka, who set up the goal for Lacazette, was a constant threat as Arteta's younger players gelled with their more experienced teammates en route to victory.
"They need the right senior players amongst them to shine. I think they showed the most important quality, which is the personality to come to any ground and play with freedom and enjoy being a footballer," Arteta said.
Having secured consecutive wins for the first time since September, the Spanish coach is hoping his side can kick on in the New Year.
"We have managed through this period to maintain our team spirit and our cohesion. Sometimes that is not easy to do, because obviously the frustration that comes out with defeats usually affects the team," Arteta said.
"Hopefully now we can go on a run of games and be more positive for that."
The win brings Arsenal up to 13th place while Brighton sink to 17th, just one place above the relegation zone.
Information from Reuters was used in this report.