After starting out at Radnički Beograd, Brzaković played for OFK Beograd in the Yugoslav Second League over the course of five seasons, before switching to Yugoslav First League side Vojvodina in the 1991 winter transfer window.
[2] He spent two years in Novi Sad before rejoining his parent club Radnički Beograd during the winter break of the 1992–93 season.
Between 1995 and 1998, Brzaković made 99 appearances for Čukarički in the First League of FR Yugoslavia.
[3] In April 1996, Brzaković was an unused substitute in FR Yugoslavia's 3–1 home victory over the Faroe Islands.
After retiring from football, Brzaković played futsal for Serbia's Marbo and Ekonomac, but also for Montenegro's Municipium.