Dragović grew up with KK Avala Ada (nowadays Mega Basket) juniors and made his debut with the senior team during the 2005–06 season.

In 2010, he signed for Spartak St. Petersburg and he stayed there till the end of 2011–12 season when his contract has expired and the team decided not to prolong it.

[6] On January 3, 2016, he left Mitteldeutscher BC and signed with Vanoli Cremona of the Italian Serie A for the rest of the season.

On October 12, Dragović signed with Nazm Avaran Sirjan of the Iranian Basketball Super League.

Dragović was also a member of the team that represented Serbia at the 2011 Summer Universiade in Shenzhen, finishing as the gold medal winners.