In a statement, a spokesperson for the minister said two asylum seekers and a staff member have contracted the disease.
None of those affected were pregnant women.
“All three people have been isolated and are receiving appropriate treatment and are expected to make a full recovery,” the spokesperson said.
“Dengue fever is not transmitted from person to person. Dengue fever is transmitted by mosquitoes. There are no drugs which can prevent dengue fever – it is controlled through minimizing exposure to mosquitoes.
“There is a comprehensive mosquito control programme in place at the offshore processing centres.”
The Refugee Action Coalition says the cases raise concerns for the welfare of asylum seekers being held on Nauru.
“This outbreak of dengue fever is only the latest reason to close the Nauru detention centre. It was always known that dengue fever was an issue on Nauru,” said spokesman Ian Rintoul.
“All the staff and the detainees suffering with dengue fever should be immediately brought to the Australian mainland.
“Arrangements must be made to evacuate all the asylum seekers from Nauru.”
Rintoul says a spokesperson for the Nauru general hospital has confirmed another 12 cases in the country.
Health officials have warned of the rise in mosquito-borne diseases in the Pacific region this year, including dengue, chikungunya and zika.