The 28-year-old arrives at the Emirates after signing a contract that runs through to 2026, with the option to extend for a further 12 months.
Arsenal will pay Brighton £21m up front for Trossard, with a further £5m coming in the form of add-ons.
The Gunners believe all the relevant documents were submitted before Friday’s midday deadline to have Trossard available for this weekend’s clash against Manchester United, but are waiting for official confirmation.
“It is good work from everyone at the club to complete the signing of Leandro,” Mikel Arteta told the club’s official website.
“He is a versatile player with high technical ability, intelligence and a great deal of experience in the Premier League and at international level.
“Leandro strengthens our squad as we head into the second part of the season and we’re all looking forward to working with him. We welcome Leandro and his family to Arsenal Football Club.”
Arsenal move quickly to target Trossard, after their top target Mykhailo Mudryk joined Chelsea for £88.5m.
The Belgian was pushing for a move away from Brighton, after refusing to sign a new deal and claiming boss Roberto De Zerbi had “humiliated him”.
As a result Trossard had been dropped from the squad for his final few weeks as Brighton player, and De Zerbi is relieved the situation has reached a conclusion.
“The situation was difficult I think, because I understood he wanted to leave,” De Zerbi told reporters.
“I’m sorry for the last period because I prefer when people are clear not when there is some different behaviours. I think anyway we are a good team and we can play well with Leo or without Leo.”