Costa made his bow for the world and European champions in Wednesday's 1-0 win over Italy, Barcelona forward Pedro scoring the only goal of the game for Vicente del Bosque's men.
The 25-year-old started his international career last year with Brazil, making two friendly appearances for the FIFA World Cup hosts.
But Costa elected to switch allegiance to Spain in October, a decision that caused Brazil coach Luiz Felipe Scolari to criticise the attacker for "turning his back" on his homeland.
However, after playing the full 90 minutes against Italy, Costa insists he feels settled in the Spain squad.
"The dream debut was to score a goal," Costa said. "It wasn't to be but I feel very happy.
"I thank my colleagues and the fans for the treatment given to me. It feels like I've been playing with them for quite some time. I felt comfortable.
"I now have to focus only on Atletico Madrid, where I have to do a good job to be able to return to the (Spain) squad.
"People know that I am Brazilian, but my desire was to play with Spain. The love given to me won't be forgotten."
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