Howard said the close relationship meant "that offers of assistance can be considered and taken without any sense that it reflects poorly on the Indonesian capacity".
There were 140 passengers and crew on board the Garuda Boeing 737, including 10 Australians and several other foreign nationals.
Australian officials are scouring hospitals to locate five of the Australians who are missing.
Indonesian and Australian crash investigators sifting through the jet's charred debris on Thursday found the aircraft's black box which will be sent to Canberra for investigations, Indonesia's Metro TV reported quoting officials in Yogyakarta.
Australia has also offered additional specialist medical care for survivors of the crash.
Canberra rushed two medical teams and police victim identification specialists to Yogyakarta after the plane careered off the runway and turned into an inferno within minutes.
On Thursday, up to 15 survivors are expected to be airlifted to the Royal Darwin Hospital in northern Australia for treatment.
"Our embassy has been in touch with the office of the Indonesian president, and we have made an offer of additional medical assistance," Howard told Australian media.
Howard was also careful not to criticise Indonesia over its aviation safety record.
"The most valuable thing I can do at the moment is to make sure that all possible help is being offered, not to make rushed judgements, to understand that countries that are not as well developed as Australia obviously have greater difficulty with infrastructure in these areas," he said.
Indonesian and Australian officials have also ruled out sabotage or any act of terror.
"We are working hard to investigate the crash, we can not even make a preliminary conclusion yet, but it is clear there are no indications of sabotage or intentional explosions in this crash as yet," Joseph Tumenggung, the head of the investigation team, told The Associated Press.