Correa agreed a five-year deal with the Spanish champions in May, but saw his proposed move from San Lorenzo delayed when a routine medical examination resulted in a heart condition being detected.
Atletico insisted they would honour their commitment to sign Correa providing he made a full recovery from surgery.
And the deal has now been completed, with Correa contracted until the end of the 2018-19 season as initially planned.
A statement on Atleti's website read: "Correa was recovering from his intervention at Majadahonda's Sports City under the supervision of the medical services of Atletico de Madrid and San Lorenzo de Almagro until he could join the group work in November.
"The U20 Argentinian international is now a part of the red and white club. He signs for the remainder of the season and four more."
