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.
"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."
