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 Australian powerhouse 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.

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 27 December 2008 Galekovic made a string of impressive saves to ensure Adelaide kept its 1–0 advantage over Perth Glory[6] and in the process giving him his fifth clean sheet of the league season.

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 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.

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.