The Argentina international has left Atletico for Mexico's Tigres this week in a permanent deal.
Correa leaves Atletico after ten years and 469 games.
"Sometimes words aren't enough... After so many years, I have to say goodbye to a place that I now feel is part of my life," he wrote on social media.
"Here I grew up, matured as a player and as a person. They taught me to defend these colors and celebrate every goal as if it were my last.
"Thank you to the fans for making me feel loved from day one. I leave with the pride of having represented you on the pitch with my heart until the last minute."
Finally, a thought for his family: "Thank you to my family, who have always been by my side. They know how much this stage has meant to me. It's not easy to say goodbye, but I can say thank you: thank you for everything, Atleti.
"We will miss you."
And Atlético responded: "It's not goodbye, it's see you later. You will always have a home here."