The Gunners said the 28-year-old had signed a long-term contract. The signing is worth £21million plus add-ons.
Trossard scored seven times in 16 Premier League matches for the Seagulls this season but his agent had requested a move after a row with Albion manager Roberto De Zerbi.
Arteta said: "It is good work from everyone at the club to complete the signing of Leandro.
"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."
De Zerbi is regretful at how the last few weeks have unfolded with Trossard, but is confident Brighton can cope without the forward.
Trossard has not played for the Seagulls since their 4-2 defeat against Arsenal on New Year's Eve.
He was dropped ahead of last weekend's match against Liverpool after leaving training early without permission.
While he has bagged seven goals this season, Brighton have won all three games without Trossard in the ranks, with last Saturday's 3-0 win over Liverpool lifting them up to seventh in the top flight.
"The situation was difficult because I understood he wanted to leave," De Zerbi told a press conference ahead of Brighton's trip to Leicester on Saturday afternoon.
"I'm sorry for the last period because I prefer when people are clear, not when there is some different behaviours.
"But we will see. I think we are a good team, we can play well with Leo or without Leo."
De Zerbi, who will be without Levi Colwill at the King Power Stadium because of a muscular problem, was non-committal when asked whether Brighton will attempt to replace Trossard in this transfer window.
"I think in the transfer market window, you always can improve, always you can buy new players," he said. "But we are at a good level now, the players in the squad are vey good players.
"It's not so easy to improve this squad. But I think (owner) Tony (Bloom) and the club if it is the possibility to improve the squad, I think we will do something."