The 27-year-old Ghana midfielder made 35 league appearances and scored three goals for Diego Simeone’s side as they finished third in La Liga last season.
He played 188 times for Atletico after joining the club in 2011.
Uruguay midfielder Lucas Torreira, 24, has moved in the opposite direction and joined the La Liga side on loan.
Arsenal have been linked with Partey, who has 29 international caps, throughout the summer transfer window.
Partey won the 2017-18 Europa League and 2018 Super Cup with Atletico and signed a new deal at the Wanda Metropolitano in 2018, until the end of June 2023.
“He is a dynamic midfielder with great energy. He brings a lot of experience from a top club that has competed at the highest level in La Liga and the Champions League for several years.
“We’re very impressed with his attitude and his approach to the game. He’s an intelligent footballer and we’re looking forward to him integrating into our system and contributing to the progress we’re building at the moment at the club.”
Arsenal were also interested in Lyon midfielder Houssem Aouar, 22 – however Aouar told French TV earlier on Monday’s deadline day that he would definitely remain at the Ligue 1 club.
Torreira – who joined Arsenal from Sampdoria for £26m in July 2018 and has made 89 appearances for the Gunners – moves to Madrid on a season-long loan.
Arsenal had already added Brazilian defender Gabriel Magalhaes for 26m euros (£23.14m) from Lille, former Chelsea forward Willian on a free transfer and re-signed Dani Ceballos on loan from Real Madrid during the summer window.
The Premier League side’s French midfielder Matteo Guendouzi, 21, has secured a loan move to Bundesliga side Hertha Berlin.
Arteta’s side have won two of their opening three Premier League games as they look to improve on last season’s eighth-place finish.