The ministry statement said: "Attempts have been made to convince them to disperse, but to no avail. The migrants' leaders resorted to incitement and attacks against the police."
Twenty-three policemen were wounded in the clashes early on Friday morning.
The clashes "resulted in chaos and a stampede, which led to the injury of 30 migrants, mostly elderly people and children, who were transferred to a nearby hospital where 10 of them died."
A Sudanese girl of about four was among the 10 killed.
A doctor at the site who examined the girl, brought to him after being found unconscious, said: "She's dead."
Reuters witnesses at Friday's scene said there were about six unconscious Sudanese, some of them children, lying on the ground after police used force to clear 3500 people who had camped in squalid conditions for more than two months.
Hundreds of Sudanese had been living in the camp since it was set up on 29 September as a protest against the UN refugee agency, whose offices are nearby.
Protesters are refusing to clear
the Cairo camp site
About 2000 riot police stormed the camp site and beat people inside with truncheons and sticks after officials failed to persuade the campers to board buses to take them to another site.
About 4000 riot police in total ringed the site in an affluent part of Cairo.
The protesters had refused to clear the area after police fired water cannons at them, demanding that the UN High Commissioner for Refugees (UNHCR) arrange for them to be flown out of Egypt.
Wilson, a Sudanese protester, said: "Let the United Nations arrange to take us to another country. Most Sudanese refugees have been subjected to violence in Egypt. We don't want to be here any more."
Puddles of blood
Puddles of blood were visible on the pavement as the Sudanese men in the camp fought back with sticks and hurled bottles at the riot police. An ambulance arrived to take away one policeman who suffered a bloody nose.
"Most Sudanese refugees have been subjected to violence in Egypt. We don't want to be here any more"
Wilson, a Sudanese protester
Antonio Guterres, the head of the UNHCR, said: "I am deeply shocked and saddened by the tragic events early today in Cairo.
"Although we still do not have all of the details or a clear picture of what transpired, violence left several people dead and injured.
"There is no justification for such violence and loss of life. This is a terrible tragedy and our condolences go to all the families of those who died and to the injured."
The UNHCR has said it is prepared to provide more assistance to Sudanese in Egypt fleeing conflict at home but cannot arrange for all of them to be resettled in another country.
Sudan's 21-year-long north-south civil war has made four million people homeless and a separate conflict in the western Darfur region has turned another two million into refugees.
A January peace agreement ended the north-south civil war but many Sudanese say it is not safe to return home as the deal is fragile.