Mornar began his career with Hajduk Split before going on to play for Eintracht Frankfurt, Sevilla, CD Ourense, Standard Liège, Anderlecht and Portsmouth.
Seemingly out of the picture at the club, he was loaned to Rennes of the French Ligue 1 for the 2004–05 season, with then Pompey manager Harry Redknapp saying that he would allow Mornar to be sold upon his return.
Upon the return of Harry Redknapp as manager in December 2005 it looked as though Mornar's contract would be allowed to expire the next Summer.
Despite his unceremonious exit from Portsmouth he enjoyed his time playing there and persuaded fellow Split native Niko Kranjčar to join the club.
Mornar made his debut for Croatia in a March 1994 friendly match away against Spain, coming on as a 46th-minute substitute for Janko Janković, but was replaced himself by Ante Miše after being on the pitch for just over half an hour.