The AFP called the attack "irresponsible", and said whoever was to blame could face prosecution.
"These attacks are irresponsible, and will not influence government policy," the organisation said in a statement.
"Activities such as hacking, creating or propagating malicious viruses are not harmless fun. They can result in serious long-term consequences for individuals, such as criminal convictions or jail time."
Police officials said the site, which hosts no sensitive information, was operating when staff left work on yesterday evening but it was down today morning.
Ties between the two countries have deteriorated after reports this week, based on documents leaked by US intelligence fugitive Edward Snowden, that Australia tried to listen to the phone calls of the Indonesian president, his wife and ministers in 2009.
The Guardian Australia reported that a member of Anonymous Indonesia, using the hashtag #IndonesianCyberArmy, had claimed responsibility for the latest attacks.
"I'm ready for this war!" the hacker reportedly said on Twitter, although AFP could not locate the tweet.
The central bank website was also targeted, with a spokesman saying it had been the subject of a denial of service attack since midnight on Tuesday -- but its back-up systems ensured that while delays may be experienced, the site remained operational.
"The bank has protections for its website, so the bank website remains secure," the spokesman said.
Anonymous is believed to be a loosely organised hacker collective that conducts online attacks internationally.