Real said on their website (www.realmadrid.com) on Tuesday that the 21-year-old had agreed a six-year contract and would be presented at the Bernabeu stadium on Wednesday.
No financial details were disclosed but Spanish media reported the transfer fee was around 30 million euros (21.1 million pounds).
Kovacic is viewed as ideal backup for Croatia team mate Luka Modric and Germany midfielder Toni Kroos at Real.
He will compete for a place with the likes of James Rodriguez, Isco, Asier Illarramendi, Casemiro and Lucas Silva although his arrival could prompt the exit of at least one of those players.
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Born in Austria, Kovacic came through the Dynamo Zagreb youth academy and helped the club win two Croatian league titles and two domestic cups before joining Inter in January 2013.
He made his Champions League debut for Zagreb against Real in 2011-12 and later that season became the second youngest goalscorer in the competition.
"With unquestionable technical ability, Kovacic stands out for his ball control, his dribbling and his passing into space," Real said.
"These attributes allow him to play as an attacking midfielder as well as a little further back where his high percentage of successful passes is a sure means of getting the side moving."
Kovacic made his debut for Croatia in a World Cup qualifier against Serbia in March 2013 and has won 21 caps including three at last year's finals in Brazil.
(Additional reporting by Zoran Milosavljevic in Belgrade, writing by Iain Rogers, editing by Tony Jimenez)
