The violence erupted in a Recife jail when an orderly protest broke down, and was brought under control only after police arrived.
One officer died of a bullet wound in hospital, while details surrounding the inmate's death were not released.
The death and injury toll was confirmed by the secretary of public safety for Pernambuco state, whose capital is Recife.
Gunfire and explosions were heard coming from inside the prison, and G1 Globo newsportal showed a helicopter with an armed official flying overhead.
Human rights NGO Conectas meanwhile has reported that Brazil prisons must boost capacity by more than 200,000 spaces to avoid overcrowding.
Pernambuco announced the opening of two new prisons in January and the expansion of another.
Less that a year ago, two prisoners were killed at another Recife prison riot.
