Human rights sources said the uprising started over the isolation of prisoners and the method of categorising inmates.
An official said the inmates died from smoke inhalation after rioters set fire to mattresses. Dozens of wounded inmates were transferred to government hospitals.
The riots began when wardens tried to move some "non-militant inmates" to other sections of the desert prison, or to Jordan's nine other detention facilities in line with a police regulation introduced last week, the official said.
It was not clear which group of prisoners started the unrest.
The regulation calls for separating convicts from detainees who are still on trial.
It also divides the prison population according to the crime they have committed.
Al-Muwaqar, to the southeast of the capital, Amman, houses hundreds of inmates, including some men convicted of involvement in al-Qaeda plots in Jordan, a close ally of the US.