The ministry, on its Facebook page, said the victim was a member of the outlawed Muslim Brotherhood arrested as he tried to plant a bomb in the capital's Al-Warraq district, without giving a date.
The suspect had insulted and provoked his police guard in hospital, threatening to kill him. The enraged policeman had shot him dead, it said.
The ministry said the policeman was arrested and an investigation opened.
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Egyptian authorities have regularly accused the Brotherhood of launching deadly attacks since the army ousted president Mohamed Morsi in 2013, a charge denied by the Islamist movement.
Tensions soared after deadly clashes between protesters and security forces in Cairo and the northern city of Alexandria on January 25 as Egypt marked the fourth anniversary of the 2011 uprising that toppled ex-strongman Hosni Mubarak.
Police arrested 516 Brotherhood supporters that day.
On Thursday, jihadists targeted security forces with rockets and a car bomb in North Sinai province, in simultaneous attacks that cost at least 30 lives.