Costa was integral to Atletico's stellar season in 2013-14, which saw them win La Liga for the first time since 1996 and reach the UEFA Champions League final, where they were beaten by city rivals Real Madrid.
The 25-year-old netted 36 goals in all competitions last term - form that earned him a spot in Spain's squad for their ill-fated FIFA World Cup defence.
It was announced on Tuesday that Chelsea had activated Brazilian-born forward Costa's release clause, leaving Atletico with little choice but to bid their talisman farewell.
However, Simeone harbours no ill-feeling towards the striker, and expects him to take the Premier League by storm.
"Diego Costa is where he wants to be and I am absolutely thrilled for him," the Argentinian told Sony One Stadium.
"I love Diego as a player - at Atletico Madrid he gave us so much and I will always be grateful to those players who have given me their all.
"Diego behaved in that way the whole time he was with us, so I wish him the very best at Chelsea, I hope he has a great season.
"He's at an age where he can continue to improve, so let's hope he can develop further.
"He'll have to adapt quickly to the system at Chelsea and to the coach, but let's hope the team can take full advantage of his characteristics.
"I see a player very much on the way up and I'm sure he'll be a success."
Costa's departure represents the continuation of a theme at Atletico, who have seen many of their best forwards leave in recent years.
But Simeone feels the club has proven itself to be more than capable of coping with the loss of stellar names.
"Top strikers have come and gone at Atletico [but] the club have always seemed to be able to minimise the impact when big players leave," he added.
"Fernando Torres left, Diego Forlan left, Sergio Aguero left, Radamel Falcao left - and now Costa and David Villa have moved on - and the team have remained competitive.
"Ateltico have always done well in that respect and I hope the same thing will happen this time with a suitable replacement for Diego."
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