It said investigations revealed the group was led by a suspect who received financial and logistical support and instructions to carry out the attacks by Brotherhood leaders residing in Qatar.
The Muslim Brotherhood had denied any involvement with the explosion at the Saint Peter and Saint Paul Church on Sunday.
The incident was the deadliest attack in recent memory on the Christian minority, who make up about 10 per cent of Egypt's population.
President Abdel Fattah al-Sisi had named Mostafa as the suicide bomber earlier on Monday, during a funeral held for the victims.
Mostafa was wanted for two other cases in connection with fundamentalist groups, the ministry said in the statement.
DNA testing of body parts found at the scene matched with his family, it said.
During investigations, authorities found two explosive belts ready to be detonated, as well as other materials used to make explosive devices, at a hideout used by Mostafa and his group.
There has been no claim of responsibility for the bombing, but Coptic Christians have been previously targeted in Egypt.
The interior ministry also named the four other people whom Sisi said had been arrested.
Rami Mohamed Abdel Hameed Abdel Ghani is suspected to have provided refuge for the suicide bomber, preparing him, and hiding the explosives.
The three others are Mohamed Hamdi Abdel Hamid Abdel Ghani, Mohsen Mostafa el-Sayed Qassem, and Ola Hussein Mohamed Ali, a woman.
The detainees will be presented to state security prosecution, while authorities are still chasing others, according to the statement.
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