After much speculation surrounding the striker’s future, Costa made the switch to Stamford Bridge after Chelsea activated his buy-out clause of £32million.
The 25-year-old spent four years with Atletico but really came to prominence last season as he helped Simeone's men to clinch the La Liga title and reach the Champions League final, netting 36 goals in all competitions along the way.
Costa has penned a lucrative five-year deal with Jose Mourinho's side and Simeone spoke of his appreciation for the Spain international's efforts during his time at the Vicente Calderon.
"I am always going to be thankful to the players that give all their best. He had excellent performances while he was with us," he said.
Atletico have consistently had to sell their top players in recent years, with Costa just the latest in a list that includes the likes of Sergio Aguero, Fernando Torres, Diego Forlan and Radamel Falcao.
However, Simeone is optimistic that another goalscorer can be unearthed to fill the boots of the man they initially signed as a back-up option from Valladolid.
He said: "Aguero left and many got scared, Falcao arrived and the team won what it won. Then Falcao left and it seemed like the team couldn’t improve. Costa arrived and the team kept on growing.
"Atletico has always handled each change very well and we expect this time to be the same."