Conte fell out with Costa in January and at the start of the summer sent the Spain international a text message to tell him that he was no longer part of his plans at Stamford Bridge.
Costa, who scored 59 goals in 120 apperances for Chelsea, refused to train at the club and spent time in his native Brazil while his entourage orchestrated the move to Atletico Madrid.
"We want to thank Diego for what he did with this club," said Conte. "We wish him all the best for the future."
Jose Mourinho was Chelsea manager when Costa joined the west Londoners in July 2014. His goals helped the club to claim the 2015 title, their first crown since 2010.
However, Costa was one of the players whose form plummeted at the start of the following season leading to the dismissal of Mourinho in December 2015.
He was reenergised under Conte and his strikes were crucial factors in the surge to the 2017 championship.
"I think now it's not important to talk about the past," Conte added. "I'm working with my players and I'm very happy to work with these players.
"Don't forget Diego and I won the title together last season. I want to thank him for his effort last season with us. Not only last season but in the period that he played for Chelsea."
Loss to League
After five games of the 2017/2018 camapign, the champions lie third - two points off pacesetters Manchester City and Manchester United. They travel to Stoke City who are managed by the former Chelsea player Mark Hughes.
"It's a loss to the Premier League," said Hughes of Costa's departure. "I used to really enjoy watching him play and how he went about his business because he was different and he had a little bit of an edge to him and it was interesting to see the reaction of other players to him.
"He's a top player and the Premier League should have the top players in world football playing in it. He's left Chelsea and the Premier League and I think it is a shame."
While Hughes's side will not have to face the menace of Costa, they will have to contend with the trickery of Eden Hazard.
The Belgian playmaker is likely to feature from the kick-off after recovering from an ankle injury sustained on international duty in the summer.
Following surgery, the 26-year-old missed the beginning of the season but after a few cameos as a substitute, he was in the starting line-up for Chelsea's League Cup win on Wednesday against Nottingham Forest.
"Eden in the last game played very well," said Conte. "I think he played with good intensity. Now his injury has been overcome. We have to play three games in a week and now there is the possibility to try to pick him for these games."