The 21-year-old has signed a five-year contract at the King Power Stadium, subject to the deal being given international clearance.
He becomes the sixth new arrival of the summer following the acquisitions of Ricardo Pereira, Jonny Evans, James Maddison, Danny Ward and Rachid Ghezzal.
Leicester boss Claude Puel said: "Filip is an exciting young player who has the potential to develop even more at Leicester City Football Club.
"He has made a big contribution to Dinamo Zagreb over the last three years and has also benefited from experiencing European football.
"I know he will give us some great options and competition in the squad and I'm delighted to have him here."
Striker Danny Ings has joined Southampton on a season-long loan which will be made permanent at the end of the 2018-19 season.
Crystal Palace have announced the signing of forward Jordan Ayew on a season-long loan from Swansea.
Tottenham have become the first team in Premier League history not to make a summer signing since the transfer window's inception in 2003