The 28-year-old has agreed a four-and-a-half-year contract at the Emirates Stadium to become Arsenal's first signing of the January transfer window.
Trossard indicated he wanted to change clubs after falling out with manager Roberto De Zerbi at the AMEX Stadium.
Arsenal turned their attentions to the Belgium international after missing out on Mykhailo Mudryk to Chelsea.
Trossard joined Brighton from Genk in the summer of 2019 and has scored seven goals in 16 Premier League games this season.
"We are really happy to have him," said manager Mikel Arteta in his pre-match press conference ahead of the game against Manchester United this weekend.
"He's a player we have followed for a while now. The opportunity came and we have the necessity to have a player in the front line that is versatile enough to play in different positions. He knows the league, he has experience here and he can have an immediate impact in the team.
"We have to think short term but also medium and long-term. He gives us all that. When you look at his career and what he's done in the last few years he's a player that fits our way of playing for sure."
Meanwhile, speaking about the Arsenal coaching team, Trossard told the club's website: "They told me they really like me as a player, that I would really suit the system and how they want to play.
"It helps me that I'm versatile and that I can play in different positions. They're looking forward to working with me and vice versa. I'm really excited to work with them and I think Mikel [Arteta] is a top coach.
"I like to be in the pocket behind the striker. On the wings, I can explore myself as well. I'm comfortable in either of those positions but if I were to choose, behind the striker - but it all depends on the system.
"This has been my best season in the Premier League until now. We did a great job at Brighton and my form has been really good. Obviously I haven't played much since the World Cup since my situation [at Brighton] but I'm so excited to start at Arsenal and show myself at a top club to help the team."
Trossard could make his debut against Manchester United on Sunday.
