The announcement came as Egypt's Christians marked Easter under tight security a week after Palm Sunday bombings claimed by the Islamic State group killed 45 people at churches in the cities of Tanta and Alexandria.
The "13 terrorist elements" belonged to cells preparing attacks against "government and Christian institutions" and police in four northern provinces including Alexandria, a ministry statement said.
It was not immediately clear when the arrests took place.
Security forces have also discovered two farms in Alexandria and the neighbouring province of Beheira that were used to make explosives and store weapons, the ministry added.
Parliament on Tuesday unanimously approved a three-month state of emergency declared by President Abdel Fattah al-Sisi in the aftermath of the attacks.
Copts, who make up about one tenth of Egypt's population of more than 92 million, have been targeted several times in recent months.
In December, an IS-claimed church bombing in Cairo killed 29 people.
The jihadist group has threatened more attacks against Egypt's Christian minority.
Last week's bombings came ahead of Catholic Pope Francis's first trip to Egypt, which a Vatican official has said will go ahead as planned on April 28 and 29 despite the attacks.
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