Elezović started playing handball for his hometown club Vrbas at the age of 19.

Over the years, Elezović established himself as one of the best players in Yugoslavia, helping his team win the championship in the 1980–81 season.

He would also play for fellow German teams TuS Hofweier and TUSEM Essen.

[1] At international level, Elezović competed for Yugoslavia in two Olympic tournaments, winning the gold medal in 1984.

Shortly prior to the 1997 World Championship, Elezović was appointed as head coach for FR Yugoslavia.