Stanić started his career with Željezničar Sarajevo. He was considered to be one of the most talented young players in former Yugoslavia. In 1992, war began and Stanić moved to Croatia, where he played for Dinamo Zagreb. After only one season, he moved to Spanish Sporting de Gijón, and year later to S.L. Benfica in Portugal. In 1995, he arrived at Club Brugge and was top scorer of the Belgian First Division that year with 20 goals. Then Parma bought him in late 1996 and he played four seasons with that side.
Stanić joined Chelsea on a £5,6 million transfer in June 2000. He became Vialli's third signing, after Jimmy Floyd Hasselbaink and Eiður Guðjohnsen. Stanić made his league debut in a 4–2 home win against West Ham United in August 2000, where he scored a double, one of them a strike from 35 yards. The goal was later nominated for the Goal of the Season on BBC's Match of the Day programme.
He won 49 international caps and scored seven goals for the Croatian national team between 1995 and 2003, making his international debut in a Euro 1996 qualifier against Estonia in September 1995. He made three appearances for Croatia at the Euro 1996 finals in England and went on to appear in all of the team's seven matches at the 1998 FIFA World Cup finals in France, where they won the bronze medal. At the latter tournament, he scored Croatia's first-ever World Cup goal in their opening 3-1 victory over Jamaica and assisted in Robert Jarni's opening goal in their 3-0 victory over Germany in the quarterfinals. Stanić also made two appearances for Croatia at the 2002 FIFA World Cup finals as a second-half substitute, only to see Croatia fall short of qualifying for the second round of the tournament. His last international appearance came in April 2003 as a half-time substitute in a friendly match against Sweden.
He was forced to retire at the age of 32 after developing a serious knee injury during the 2003-04 league season.