The defending champions crashed out on Wednesday, going down 2-0 to Chile at the Maracana to confirm their elimination.
The defeat sparked suggestions among the media that the Spanish team - winner of the last three major international tournaments - had reached the end of an era.
After the game, Alonso questioned the squad's hunger and ambition, remarks that have not gone down well with team-mate Costa.
"I don't agree with his comments," he told Marca.com.
"Everybody here was focused on winning.
"I'm here because I wanted to win, I wanted to reach the final and do a good job, but it wasn't to be.
"We obviously lacked something, but in terms of quality we're superior to many other teams."
Before flying home, Spain take on Australia in Curitiba on Monday, with both sides looking to claim their first win in Brazil and end disappointing campaigns on a high note.
