Security forces raided a group hideout, where members "held organisational meetings and made explosive devices to be used in a series of hostile operations", in the October 6 district of western Cairo.
Hassam claimed responsibility for the December 11 bomb attack inside a Cairo church that killed 26 people and a December 9 bombing that killed six policemen in western Cairo.
The ministry said its forces responded to gunfire from inside the apartment, while a police official said on condition of anonymity that the incident took place on Saturday.
Hassam, in a statement circulated on social media, said one of its members was killed, identifying him as Mohamed Ashour Desheisha.
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Attacks on security forces have escalated since the military's July 2013 ouster of Egypt's former Islamist president Mohamed Morsi, who hails from the Muslim Brotherhood group, and a crackdown on Islamists.
Analysts had warned that Islamists affiliated with the Brotherhood, though not necessarily under their control, could step up violence in the face of the crackdown.
Militants in North Sinai province, at the heart of an insurgency against security forces, pledged allegiance to the Islamic State jihadist group in November 2014.
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