Barisic began his career in 2002 with the ACTAS before moving to Canberra Deakin in July 2003.
In January 2004 he signed his first professional contract with Melbourne Knights, then left in the summer of 2006 for Europe and signed a contract for Germany-based Oberliga Nord club, Arminia Hannover.
On 13 December 2007 he signed with his former club Melbourne Knights in the Victorian Premier League. He impressed regularly, becoming a key member of their squad and scoring 19 goals in 21 games in the 2008 season.
On 10 December 2008 he was signed by Gold Coast United on a one-year contract. At Gold Coast United he made six appearances and scored one goal in his first season there. He was signed on for one further year. In his second season at GCU, he made 11 appearances and again scored one goal.
After leaving Gold Coast, "Bara" signed for Indonesian side Persebaya Surabaya in the Indonesian Premier League. He did very well there, scoring 11 in 14 games.
He then signed for another Indonesia Super League side, this time Arema Cronus F.C., where he managed 8 goals in 15 appearances.
Andrew officially signed for East Bengal on 10 January 2013. He then made his debut for East Bengal on 25 January 2013, coming on as a 59th-minute substitute for Manandeep Singh, against Pune F.C. at the Balewadi Sports Complex in which East Bengal won 2–1. On 3 April 2013, he scored brace in 2013 AFC Cup against Singaporean opponent Tampines Rovers in group stage match. He again scored in same competition against a Vietnamese opponent Xuan Thanh Saigon in last group stage match, after winning that match East Bengal tops the group and qualified for the knockout stage of AFC Cup. On 8 May, he scored his 1st goal in the 2012–13 I-League in a 6–0 win over United Sikkim F.C. He ended his one-year stint with East Bengal with 15 goals in 22 matches.
He returned to Melbourne Knights for their 2013 VPL campaign, while the majority of the rest of Asia was in off-season. He managed 7 goals in 8 games in his third stint with the Knights.
On 31 December 2013, Barisic signed for South China. At South China, Barisic managed 13 goals in 18 appearances, 6 in the Hong Kong PL, 2 in AFC Qualification and 1 in the Senior Challenge Shield.
Barisic then signed for newly formed Kerala Blasters FC in the newly formed Indian Super League. In the ISL side, Barisic managed 11 appearances in which Kerala came 2nd.
Barisic returned for the 4th time to Melbourne Knights in January 2015 scoring 12 goals in 16 games.
On 1 July 2015 Andrew Signed with Hong Kong Premier League Side Eastern.As of 9 May 2016