Attacks on the security forces have become commonplace since the army deposed president Mohamed Morsi of the Muslim Brotherhood last July after mass protests against his rule. The state has declared the Brotherhood a terrorist group, but its leaders say it remains committed to peaceful activism.
In a separate incident on Friday, the army said its special forces had killed a top Sinai militant, naming him as Nour al-Hamdeen. A brief statement posted on the army's Facebook page did not say to which group he belonged.
Egypt's Interior Ministry said the Nile Delta shootout began when Brotherhood activists on motorbikes were spotted attempting to set alight a police checkpoint on a road between the cities of Tanta and al-Mahalla al-Kubra north of Cairo.
They opened fire on police as they tried to flee, it said.
"The forces encircled them and took the necessary security measures and exchanged fire. That led to the death of two of them," the ministry said in a statement. Three others were arrested, it added.
The Brotherhood had no immediate comment on the incident.
The death in Alexandria was confirmed by senior ambulance service official Omar Nasr. The website of the Brotherhood's Freedom and Justice party identified the man killed as 50-year old Abdel Hakim el-Zamzami. It said he had been shot by security forces and "militias" as they tried to break up a protest.
The increased political violence in the Arab world's most populous country poses a big challenge to former army chief Abdel Fattah al-Sisi, who toppled Morsi last July and is expected to become Egypt's next president after a May election.
Ansar Bayt al-Maqdis
Nearly 500 people, the bulk of them soldiers and policemen, have been killed in militant attacks since last June, the government said last month.
On Thursday, a radical Islamist group, Ansar Bayt al-Maqdis, released a video identifying the bomber behind a Dec. 24 suicide attack on a police compound in the Nile Delta as someone who was shot during Cairo protests against Morsi's overthrow last year.
It is the first time the group has identified one of its bombers as someone who took part in those pro-Morsi protests.
Ansar Bayt al-Maqdis, or Supporters of Jerusalem, has claimed responsibility for some of the most devastating attacks, including suicide bombings. The United States this week designated it as a terrorist organisation.
The video names the Dec. 24 bomber as Abu Maryam Imam Marai and says he was shot during Cairo protests last year. A young man with a tightly cropped beard, he goes on to say that members of the security forces who kill Muslims should be killed.
It also shows two men, their heads covered with scarves, wiring together what appear to be barrels of explosives ahead of the attack, in which at least 16 people were killed.
"The path to establishing religion is not by preaching alone, as people claim, but by preaching and jihad," the bomber says, flanked by two semi-automatic rifles.
The group emerged in 2011 in North Sinai, where the army is waging a campaign to reassert government control.
By contrast, the Muslim Brotherhood insists it remains committed to peaceful change, a message reiterated this week by a statement from abroad by its secretary-general Mahmoud Hussein, one of the few leaders of the group not now in jail.
Some analysts question how much control the Brotherhood's leadership now wield over the youth base of a movement that has been outlawed and driven underground.
The security forces killed hundreds of Morsi supporters in the weeks after his removal from power, culminating in the violent dispersal of two pro-Morsi sit-ins in Cairo on Aug. 14.