Zelic started his career in the old Australian National Soccer League, where he played with clubs Sydney Croatia and Sydney Olympic. In his junior years he played for Croatia Deakin.
Zelic spent most of the 1990s playing in Europe, most notably for Borussia Dortmund where he reached the UEFA Cup Final in his first season with the club, losing to Juventus Turin of Italy and won the German Bundesliga title in 1995. He had a small stint in the Premier League for QPR, moving for a club record £1.25 million, which was beset by injury and discontent. After just 11 Premier League games for the West London club, he departed to Eintracht Frankfurt in Germany.
After only a few months he signed for AJ Auxerre of France. At Auxerre, he replaced Laurent Blanc who moved to FC Barcelona, and participated in the UEFA Champions League. In July 1996, Zelic was selected to play for a World All Star team vs. Brazil at Giants Stadium, New York. The All Star team was coached by Euro 92 winner with Denmark, Richard Moller Nielsen and consisted of such players as Jürgen Klinsmann, George Weah, Marcel Desailly, Lothar Matthaus, David Ginola and Fernando Redondo. In 1998, Zelic decided to move back to Germany and went on to make 102 appearances for 1860 Munich in the Bundesliga.
In 2002, he moved to Japan to play in the J-League where he stayed for two years and played under Dutch coaches Hans Ooft and Wim Jansen, winning the Japanese Cup with his club Urawa Red Diamonds. Zelic signed for FC Wacker Tirol of Austria in July 2004 and stayed there for one season.