Bozanic was born in Sydney but started his professional career in England before returning to play for Central Coast Mariners in the A-League.

He is the son of former Socceroo Vic Bozanic who was part of the West Adelaide team which won the 1978 National Soccer League.

[11] Shortly after, on 26 November 2009, Bozanic joined former Reading assistant manager Kevin Dillon at League Two team Aldershot Town on a one-month loan deal.

[14] After being released by Reading at the end of the 2009–10 season,[15] Bozanic returned to Australia to rejoin Central Coast Mariners.

[20] Throughout February, Bozanic was not featured at the club, due to international commitments for the national team at the qualifying campaign for London 2012, where he was captain, but it was successful.

[25] After three and a half week out, Bozanic made his return, where he set up the only goal for Bernie Ibini-Isei, in a 1–0 win against Perth Glory.

In June 2013, Bozanic was among three players to be linked with Asian clubs, having been attracted by South Korean sides Busan IPark and Pohang Steelers.

[28] On 14 June 2013, Bozanic signed a two-year deal to join Swiss Super League side Luzern.

[29] Thirty days later, Bozanic made his debut, in the opening game of the season, in a 2–0 win over Lausanne and scored his first goals in the next match, in a 4–2 loss against Aarau.

[33] He made his competitive debut 19 days later in a 3–1 win against Adelaide United in the 2015 FFA Cup Quarter Final, where he was involved in setting up his team's second goal.

[40] On 18 December 2021, Bozanic played his 100th A-League Men game for the Mariners, scoring a late free kick in a 2–0 win over Western Sydney Wanderers.

[44] Bozanic made his debut for the Wanderers on 10 December 2022 against Wellington Phoenix, and scored his first goal for the club against Brisbane Roar in his third appearance.

After 10 months as a free agent following his release from the Western Sydney Wanderers, Bozanic announced his retirement from football on his Instagram page on 17 November 2024.

He was a member of all the lead-up squads prior for the 2009 FIFA U-20 World Cup, but missed out on his place at the tournament due to injury.

[52] Bozanic was selected in the Australian squad for the 2014 FIFA World Cup, and played in Group Stage matches against Netherlands and Spain.