He played junior football in Australia before joining English side Liverpool as a youth player, where he appeared for the Under 18 and reserves sides. After a loan move to Accrington Stanley he joined Oldham Athletic, before brief stays at Aris FC and Carlisle United. In 2014, he returned to Australia to play for the Wanderers in the A-League.
He has been capped by both Australia and Greece at youth level.
Born in Sydney, New South Wales, Bouzanis was raised in Australia to Greek parents, and attended De La Salle College in Cronulla. His talents were first seen at his local soccer club, Lilli Pilli FC, whilst playing in the Under 6's, before moving to Carrs Park FC, alongside close friend Nikolas Tsattalios. Bouzanis then went on to play in the NSW State League for St George Saints at 14 before joining the NSW Institute of Sport.
Bouzanis spent three weeks with the Liverpool first team squad in January 2007 and was offered a three-year deal. Liverpool's manager Rafael Benítez stated that he rated the Australian as "the best goalkeeper in the world for his age."
He was loaned to A-League side Sydney FC as understudy to Clint Bolton and was included in the club's squad for their AFC Champions League 2007 campaign, but did not appear at the tournament, and was subsequently delisted.
He appeared for Liverpool's under 18 and reserve teams and after a loan deal with Conference National team Wrexham fell through in July 2009, in November he joined Accrington Stanley on loan, subsequently making 14 league appearances for the club.
In May 2011 Bouzanis left Liverpool after mutually agreeing to a release with one year remaining on his contract.
In July 2011, Bouzanis claimed he had agreed terms with A-league club Melbourne Victory but this was denied by the club.
On 26 November 2011, Bouzanis joined Football League One side Oldham Athletic on non-contract terms as cover for injured back-up keeper Paul Gerrard following a four-week trial. His time at the club included an FA Cup tie against former club Liverpool at Anfield, although Bouzanis spent the match on the bench as an unused substitute. The following month, Bouzanis signed a new deal with the club, until the end of the season.
He made his competitive Latics debut on 27 March 2012 coming on as second-half substitute against Leyton Orient. After a spell in the side at the end of the 2011-12 season when Oldham's first choice keeper Alex Cisak was injured. At the end of the 2011-12 season, Bounzanis signed a new one-year contract following his good display for his performance.
At the start of the 2012-13 season, Bouzanis suffered an ankle injury during the pre-season friendly. He dropped again to the bench at the beginning of the 2012-13 season, before regaining his first-team spot in September. On 27 January 2013, Bouzanis played in Oldham's upset 3-2 win in a F.A. Cup club 4th round tie against Liverpool.
After beating Liverpool, Bouzanis' Oldham faced Everton. In this game Bouzanis came under criticism after going up for a last minute corner which Matt Smith scored from. He was criticised for jumping and appearing to handle the ball. Video replays confirmed he did not handle but many pundits said that this was unsportsmanlike. Bouzanis was subsequently released by Oldham at the end of the 2012-13 season.
Following his departure from Oldham Athletic, Bouzanis joined Greek side Aris FC but failed to make a senior appearance.
Bouzanis was released by Aris FC in December 2013, and joined Carlisle United on a free transfer on 28 January 2014, on a short-term deal until April, but made no appearances.
In May 2014, Bouzanis returned to his home town, signing with Western Sydney Wanderers.
Bouzanis replaced Tando Velaphi at Melbourne City during the January transfer window of the 2015-16 A-League. He signed a one-year contract extension on 28 April 2016, with his new deal to end at the end of the 2016-17 A-League season. Bouzanis made his competitive City debut on 8 October 2016 in the club's first game of the 2016–17 A-League, coming on in the second half after Thomas Sørensen was sent off in a 1–0 win. On 8 February 2017, Bouzanis was ban for five games for using a racial slur against Melbourne Victory, Besart Berisha after appearing to call the Albanian striker a "gypsy".
Bouzanis was first named in the Australian U17 side in 2005 at the age of 15. He was also later included in the squad for Australia's unsuccessful 2006 AFC U17 Championships qualifying campaign.
On 2 February 2008, Bouzanis was invited by the Greek FA manager Alexis Alexiou to represent the youth team against Slovakia on 5 February 2008. Bouzanis visited Greece on 3 February 2008 for an interview with Greek radio station SportFM, and also to play in the friendly against Slovakia, in which he kept a clean sheet. He also played in a 1–0 win against Serbia in March 2008.
Despite playing for the Greek U19 team Bouzanis remained eligible to play for either the Greek or Australian full international sides. The Australian press stated that his departure would be a "disaster" likening it to the loss of Joey Didulica to the Croatian national side.
In April 2008 he was called up to the Australia Under 23 squad to compete in the Intercontinental Cup in Malaysia as part of their preparations for the Olympics due to Danny Vukovic's suspension. He played in the 4–0 defeat of Togo in the final group match. With the announcement that Vukovic's suspension would not be lifted for the Olympics, suggestions were made that Bouzanis would be included in the squad but this did not eventuate.
In June 2008, Bouzanis accepted an invitation to represent Greece in the UEFA U-19 European Championships. and stated he was proud to be competing at the finals with Greece. Bouzanis' decision to play for the Greek under-age team upset some of Australia's football community. Bouzanis' agent suggested he still wanted to play for Australia. In April 2008 Australian national coach Pim Verbeek met Bouzanis in Liverpool to discuss his football allegiance.
In February 2009, his agent stated that Bouzanis wanted to play for Australia rather than Greece. Bouzanis subsequently appeared in Australia's first two group matches at the 2009 FIFA U-20 World Cup in Egypt. Bouzanis' next international appearance was as a half-time substitute in an Under-23 friendly on 1 June 2011 against Japan. He was subsequently named in the squad for the Olyroos London Olympics 2012 qualifying matches against Yemen in June 2011.
Statistics accurate as of match played 11 April 2015Western Sydney Wanderers
AFC Champions League: 2014