Militants detonated an Improvised Explosive Device, destroying three armoured vehicles and a signal-jamming vehicle.
The security forces broke into an apartment in one of the neighborhood's residence building, where the suspects were hiding.
The officials spoke on condition of anonymity because they were not authorized to brief media.
At least 18 police forces, including a general, have been killed in Egypt's Sinai Peninsula after militants ambushed their convoy, in one of the deadliest terrorist attacks in the restive region since January.
Islamic State of Iraq and the Levant (ISIL) claimed responsibility for the attack in a brief statement carried by its Aamaq website. The Egyptian interior ministry confirmed the attack in a statement, reported Egyptian news daily Al-Ahram. It did not provide any casualty figures. The Muslim Brotherhood denies this.
Egypt has been suffering from a spike in terrorist attacks targeting civilians in the mainland over the past year.
Monday's attack came a day after authorities said they had busted an extremist cell planning attacks in Cairo.
The exchange of gunfire took place in the densely populated Cairo district of Ard el-Liwa and wounded five policemen, including two officers, said the officials. They said the militants sneaked into the capital from northern Sinai.
The violence surged after the ouster of Islamist President Mohamed Mursi in 2013, which was followed by a crackdown on the main Muslim Brotherhood group that has killed hundreds and led to the jailing of thousands more.
The attack occurred near the town of El-Arish, administrative centre of province Northern Sinai.