The 21-year-old will cost Rovers an initial £6million after the Ewood Park club agreed a fee with the Croatian's former employers Hajduk Split.
The Croatia international has reportedly agreed a four-year contract with Blackburn, who are expecting big things from him.
Sam Allardyce said: "He has a terrific goalscoring record. He is only 21 and scores one goal every 1.6 games. That is why we have forked out as much money as we have."