Born in the city of Nikšić (former Yugoslavia, now Montenegro), Damjanović started his professional career with local club FK Čelik Nikšić in 2009. After a few years of playing club football in Montenegro, Slavko moved to Serbia with FK Spartak Subotica and a year later joined FK Bačka 1901.
Over the next few years, Slavko played club football for clubs in Hungary, South Africa and Uzbekistan and also had brief intervals of playing club football back home in Serbia & Montenegro. The 28-year-old left-footed 6’2″ center back joins Chennaiyin FC from Serbian club FK TSC Bačka Topola on a free transfer.
During the course of his career, Damjanović has made over 150 club appearances and has won the 2016-17 Montenegrin Cup with FK Sutjeska Niksic, the 2019-20 Montenegrin League with Buducnost Podgorica and the 2017-18 South African Cup with Bidvest Wits FC. Slavko has also featured in the UEFA Europa League qualifiers with Buducnost Podgorica.