The action claims the men's lives "are in peril" due to "understaffing and neglect" in Mississippi's prisons.
Jay-Z launched the action through Team Roc, the philanthropic division of his entertainment empire Roc Nation.
It comes after five prisoners were killed in attacks in the state's prisons in one week over the New Year.
The lawsuit alleges chronic underfunding and understaffing has resulted in "prisons where violence reigns" and an "unthinkable" spate of deaths.
'Living in squalor'
"The underfunding also forces people held in Mississippi's prison to live in squalor, endangering their physical and mental health," the action continues.
It claims the prison at Parchman - where three of the five recent victims died - is subject to flooding, black mould and an infestation of mice and rats.
The lawsuit seeks damages for the inmates and an order forcing the Mississippi Department of Corrections to take action.
The case was filed on Tuesday against outgoing MDOC commissioner Pelicia Hall and Marshal Turner, superintendent of the Mississippi State Penitentiary at Parchman.
A spokesperson for the MDOC said it "does not discuss pending litigation".