Brendan Rodgers conceded defeat in his pursuit of Costa over the weekend after learning that Costa had agreed a new deal to stay in the Spanish capital.
And Atletico have now confirmed that he has penned a three-year extension to keep him at the club until 2018.
Liverpool had reportedly tried to activate the buy-out clause in Costa's contract and other clubs were believed to be monitoring his situation, but he is thrilled to have committed his future to Atletico.
The Brazilian, who scored 20 goals last season, spent a couple of years with Atletico early in his career before returning to the club from Real Valladolid in 2010.
"I am very happy with this deal after fighting for a long time to win the affection of the fans, my team-mates and people inside the club," he told the club's official website.
"It is a very special moment and I am very happy here."
Sports director Jose Luis Caminero said: "Costa is one of the key pieces in the team and it continues our policy of trying to maintain the group that have brought us so much success in recent years. Extending his contract was one of our objectives."