The Uruguayan joins the Spanish giants on an initial loan deal, which includes an option to buy next summer, believed to be for a fee of £22 million.
Torreira joined the Gunners from Sampdoria in a deal worth £26 million last summer and shone in his first season at the Emirates.
But, last season saw the 24-year-old used out of position by Unai Emery before his December sacking and then he lost his place in the team under Mikel Arteta at the back end of the campaign.
The Gunners have been looking to move Torreira on all summer, with a number of Serie A clubs interested in his signature, but Diego Simeone convinced the midfielder to move to the Wanda Metropolitano.
"Everyone at Arsenal wishes Lucas well during his time with Atletico Madrid this season," said the club in an official announcement.