An officer also died in a shootout in the southern province of Qena on Tuesday, security officials said.
Egypt is battling a local affiliate of the Islamic State jihadist group, which has claimed attacks that have killed more than 100 Copts since December.
The shootout occurred after a suspected jihadist, who had previously been detained, guided police to an alleged hideout in Qena, the ministry said in a statement.
"As soon as the security forces reached the location, the terrorist elements suddenly opened fire using all types of weapons, which forced them to retaliate," it said.
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At the hideout, police found weapons and "gold jewellery which was probably stolen from some of the Christian victims" of a previous attack.
On May 26, masked gunmen killed 29 Copts as they travelled in a bus to Saint Samuel monastery in Minya province south of the Egyptian capital.
The bus attack followed two suicide bombings of churches in April that killed 45 Copts. In December, a suicide bomber struck a church in Cairo, killing 29 Copts.
Egypt's IS affiliate is based in North Sinai province, where hundreds of soldiers and policemen have been killed in attacks since the military's ouster in 2013 of Islamist president Mohamed Morsi.