Police said about 90 people, including 20 police officers, were wounded on Friday outside St George Church in the Mediterranean city in what began as a demonstration against a theatrical play deemed offensive to Islam.
The rioters hurled stones that injured police officers, smashed church windows and damaged the white facade of the three-storey church, a police official said.
They also set a police car on fire and wrecked eight other cars, the Interior Ministry said in a statement.
One of the victims, Mohammad Zakariya, died late on Friday of his injuries in hospital, the ministry said. Two others died shortly after midnight after the police opened fire on the protesters.
The Interior Ministry described the protesters as "fanatic elements" who "escalated a negative reaction to a play". The ministry said about 5000 Muslims had marched to the church after Friday noon prayers at the local mosques.
The riot was the latest in a series of incidents at the church in which Muslims have condemned the distribution of a DVD of a play, performed at the church in 2003, that tells the story of a young Christian who converts to Islam and becomes disillusioned.
Coptic Christians make up about
10% of Egypt's 70m population
On Wednesday, a Muslim man stabbed and seriously wounded a novitiate nun inside the church. He was arrested.
The church's director, Father Augustinous, said it was difficult to explain the reaction to a one-time performance that took place two years ago.
"There are so many questions on what is behind all of that," Augustinous said.
Muslims who have seen the DVD say the play is insulting.
The Interior Ministry, which is responsible for security in Egypt, has dispatched thousands of police to Alexandria in anticipation of sectarian trouble.
Coptic Christians make up about 10% of Egypt's overwhelmingly Muslim population of 70 million.
Accusations of forced conversion are common.