Galekovic's junior football was mostly played at Chelsea Hajduk in Melbourne's south-eastern suburbs.
Galekovic played school football for St Bede's College.
Galekovic started his career with Morwell based National Soccer League (NSL) side Gippsland Falcons as a 19-year-old in December 2000 before signing with South Melbourne the following season. He stayed with South Melbourne until the NSL's closure in 2003–04 where at season's end he secured a place on the list of Beira Mar in Portugal on the back of holding the number one spot for Australia's Olympic football campaign in Athens in 2004.
The period at Beira Mar was generally unsuccessful for Galekovic, with him seeing little first team action and thus, he returned to Melbourne to sign for the city's inaugural A-League side Melbourne Victory for season 2005–06, debuting in the opening match against Sydney FC.
Galekovic played 11 matches for the season in an unusual goalkeeper rotation policy employed by Melbourne manager Ernie Merrick in which the other 10 matches saw teammate Michael Theoklitos in goal. Galekovic was second choice keeper for the Victory in the 2006–07 season but an injury to Theoklitos gave Galekovic his first start of the season in the round 13 clash against Queensland Roar at Suncorp Stadium.
On 30 October, Eugene signed for Melbourne Victory rivals Adelaide United as a replacement for injured keeper Daniel Beltrame and made his debut against the Queensland Roar at Hindmarsh Stadium. On 20 January 2008, Galekovic kept his first clean sheet for Adelaide in a 2–0 win over Brisbane at Hindmarsh Stadium. On 27 December 2008 Galekovic made a string of impressive saves to ensure Adelaide kept its 1–0 advantage over Perth Glory and in the process giving him his fifth clean sheet of the league season.
In February 2010, Galekovic capped off another brilliant year by winning the Adelaide United Player of the Year Award.
In the middle of the 2011–12 A-League Jon McKain resigned his role as captain at Adelaide United. It was then announced that Eugene Galekovic became captain.
On 11 March 2016 Galekovic reached 250 appearances despite a 1–0 defeat to Melbourne City FC.
On 16 June 2017 Galekovic signed a two-year deal with Melbourne City FC.
Galekovic has been called up for Australia in a preliminary squad with teammates Scott Jamieson and Paul Reid after impressive performances in the A-League and the Asian Champions League. He made his first senior international football debut for the Socceroos on 28 January 2009 in an AFC Asian Cup qualifying match versus Indonesia. He was disappointing in his last outing against Kuwait, by letting in two poor goals after Australia conceded a 2–0 advantage. After Brad Jones left the 2010 World Cup 23-man squad to help his son battle leukaemia, he was called in as a replacement, after missing selection in the 23-man squad first time around. Eugene was selected by Holger Osieck for a 2013 EAFF East Asian Cup qualifying tournament. He started the first match against hosts Hong Kong and came up with audacious stops to save Australia from humiliation as they won 1–0 with a solo effort from Brett Emerton. The other game that Galekovic was a part of a 9–0 thrashing of Guam. After a string good performances for Adelaide United, Holger Osieck selected Eugene Galekovic as part of his 23-man squad for the decisive 2014 FIFA World Cup qualification (AFC) match against Oman.
He was part of the Australia squad for the 2014 World Cup, but did not play.As of 9 May 2016
CS = Clean sheets
1 – includes A-League final series statisticsMelbourne Victory:
A-League Championship: 2006–07
A-League Premiership: 2006–07
FFA Cup: 2014
A-League Premiership: 2015–16
A-League Championship: 2015–16
2 – includes FIFA Club World Cup statistics; AFC Champions League statistics are included in season commencing after group stages (i.e. 2008 ACL in 2008–09 A-League season etc.)
AustraliaAFC Asian Cup: 2015
A-League Goalkeeper of the Year: 2008–09, 2009–10, 2013–14, 2014–15
Adelaide United Club Champion: 2008–2009, 2009–10
Adelaide United Player's Player of the Year: 2008–2009, 2009–10, 2011–12
A-League PFA Team of the Season: 2008–09, 2009–10, 2012–13, 2013–14, 2014–15
A-League All Stars: 2014
PFA A-League Team of the Decade: 2005–15
Individual RecordsMost Goalkeeper of the Year Awards: 4