He played for the Penrith Panthers in the New South Wales Rugby League competition, primarily as a second-row, but also as a prop or lock.
A Wentworthville Magpies junior, Zivanovic was a tireless worker in both attack and defence in his seven years at the Penrith club, often topping the tackling tables for his side.
Zivanovic played in Penrith's first ever finals campaign in 1985 and featured in both matches against Manly and Parramatta.
On 6 March 2017, it was revealed that then Penrith player Bryce Cartwright was involved in an affair with another woman and had got her pregnant.
Cartwright reportedly sent text messages to the woman when he discovered she was pregnant saying “You do realise there is no way I’ll be at the birth, We won’t keep in contact at all".