The fatalities in the attack near the Khan el-Khalili bazaar in the heart of Cairo were all policemen. The explosion late Monday also wounded two other policemen and a woman, officials said.
The attack was a rarity for the central area of Egypt's capital amid a years-long security crackdown under President Abdel-Fattah el-Sissi.
The Interior Ministry said the attacker, 37-year-old al-Hassan Abdullah, blew himself up after police officers approached to arrest him. He was wanted in a bombing last Friday near a mosque in Cairo's district of Giza and the police had been monitoring his movements, the statement said. The attacker's affiliation was not known and no militant group claimed responsibility for the bombings.
The ministry had blamed members of the outlawed Muslim Brotherhood for last week's attack, which it said targeted a security checkpoint and wounded three people.
Following Monday's explosion, which shattered windows and blew curtains off nearby balconies, police and soldiers cordoned off the narrow streets around the bazaar. A body, presumably of the attacker, covered with a white sheet stained with blood, was seen lying on the ground in the blocked-off area, close to Egypt's renowned Al-Azhar mosque.
In a house nearby, police found a bomb and bomb-making material, which prompted the evacuation of the whole building, said the security officials.
In the restive north of Egypt's Sinai Peninsula, security forces killed 16 militants and seized explosives and weapons in two raids, security officials said Tuesday. The raids involved clashes with Islamic militants in the desert outside the city of el-Arish, they said. It was unclear when the battle took place.
Officials spoke on condition of anonymity because they were not authorized to talk to the media.
Egypt has been battling Islamic militants for years, but the insurgency gained strength after the 2013 overthrow of an elected but divisive Islamist president, Mohammed Morsi. The militants have mainly targeted security forces and Christians.
Egypt last year launched a wide-scale security operation focused on northern Sinai, where an Islamic State affiliate has carried out many attacks in recent years.