A senior Indonesia MP branded the decision as an "unfriendly" act on Australia's part.
A spokesman for Indonesian President Susilo Bambang Yudhoyono said Jakarta is drafting a response to the announcement.
The group of 43 West Papuans landed at Cape York in January, saying they feared being executed if they returned home, however Indonesian officials have denied this charge.
Immigration Minister Amanda Vanstone on Thursday said 42 of the group have received temporary protection visas and will be relocated from Christmas Island to Melbourne.
"They will have access to the same services as other TPV holders," Senator Vanstone said.
She said a decision is still pending on one of the asylum seekers as there are further, specific case issues to be addressed.
Genocide claims
The 36 adults and seven children, who spent five days at sea before arriving on Cape York, have accused the Indonesian military of conducting genocide in their homeland.
Indonesian President Susilo Bambang Yudhoyono earlier phoned Australian Prime Minister John Howard, saying the group should not be given political asylum and should be returned to Indonesia.
He gave an assurance they would not be prosecuted.
The families have been staying in immigration housing on Christmas Island while the single men have been kept in the detention centre.
Refugee groups have called for them to be released on TPVs since their arrival.
"Unfriendly gesture"
Nationalist Indonesian MP and member of the country's foreign affairs commission, Djoko Susilo, said the decision to allow the Papuans to remain in Australia was "too much".
"This is an unfriendly gesture by the Australian government," he said, adding that the Indonesian government must mount a diplomatic protest.
Australian Foreign Minister Alexander Downer personally informed his Indonesian counterpart of the decision by telephone, and a spokesman said he did this as "it's a matter of some significance between our two countries".
Senator Vanstone said each claim was considered on its individual merits.
She would not be drawn on whether the genocide claims had influenced the decision.
Ten of the 42 detainees are currently receiving medical treatment in Perth and all of them will soon be relocated to Melbourne.
The remaining asylum seeker in still on Christmas Island, and will remain there until other issues are considered, said Senator Vanstone.