Kovacevic was born in Kovin, Serbia, on November 18, 1973. He played for Proleter Zrenjanin and Red Star Belgrade in Serbia before moving in 1996 to Sheffield Wednesday. Later that same season he transferred to Real Sociedad where he established himself as a footballer. In 1999 he had a 2-year unsuccessful spell in Italy with Juventus and later Lazio.
Kovacevic has been capped by Yugoslavia and was a member of their 1998 World Cup squad.
In 2008, doctors for Kovacevic detected high blood pressure and a narrowing of the arteries. On January 13, 2009, he underwent heart surgery (coronary angioplasty) in San Sebastian, Spain. The operation was deemed a success but the doctors recommended that he end his career at this point.