The ministry said on Friday the Nile Delta shoot-out 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, the ministry 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's statement added. Three others were arrested, it said.
The Brotherhood had no immediate comment on the incident. The death in Alexandria was confirmed by senior ambulance service official Omar Nasr, according to the Reuters news agency.
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.
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.
Nearly 500 people, the bulk of them soldiers and policemen, have been killed in attacks since last June, the government said last month.
In a separate incident on Friday, the army said its special forces had killed a top Sinai fighter, naming him as Nour
al-Hamdeen, according to Reuters. A brief statement posted on the army's Facebook page did not say to which group he belonged.