Bookmakers' odds for the Balkan nation to win the tournament have been changed from 16-1 to 8-1 after the 2-1 victory in Bordeaux. Kalinic scored the equaliser to cancel out Spain's early goal before Ivan Perisic grabbed a late winner.
"It was difficult after we fell behind but we showed that we are shoulder to shoulder with the world's top teams," Kalinic said. "We are now one of the big favourites of this tournament."
Having been fined 100,000 euros for crowd trouble caused by their fans in a 2-2 draw against the Czechs, a makeshift Croatian side came from behind against the Spaniards after coach Ante Cacic made five changes to his side.
Kalinic's confidence may not be misplaced as Croatia's 20-year-olds Tin Jedvaj and Marko Rog, both of whom were winning only their third caps, showed maturity beyond their years.
Forward Marko Pjaca also shone as he delivered a defence-splitting pass for Perisic's winner, with injured playmaker Luka Modric, Croatia's key player, celebrating on the bench.
But Ivan Rakitic, who got the better of his Barcelona team mates featuring for Spain, struck a note of caution.
"Let's not get carried away," said the attacking midfielder who added he had "a few things to explain" to his Spanish wife.
"We are overjoyed, we will celebrate this win thoroughly but there is no room for complacency."
Perisic added: "We bounced back after what happened against the Czechs and won the group, but it will mean nothing if we fail now."
(Writing by Zoran Milosavljevic; Editing by Adrian Warner)
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