Currently he is the head coach of Jugović member of Serbian Handball Super League.
After starting out at Vrbas, Elezović played for Sintelon and Partizan, before moving abroad.
He would go on to play in Slovenia, France, Switzerland, Qatar, Israel and Hungary.
With Vojvodina, Elezović won three consecutive national championships (2012–13, 2013–14 and 2014–15).
At international level, Elezović represented Serbia at the 2016 European Championship.