Campaigners have called for a public inquiry in Northern Ireland following two reports in the Irish Republic which said mistreatment in Catholic and state-run institutions was endemic.
Peter Robinson and Martin McGuinness saw a delegation representing victims at Stormont.
One, John Meehan, said: "For someone who never walked out the front gate of an orphanage, always walking out the back gate, for me to be here today is a big apology."
The Northern Ireland Assembly has already voted for an inquiry into institutional abuse in NI. The victims want to see an inquiry, damages and redress, extra support and an apology.
Mr Meehan, 63, lives in Athlone, Co Offaly, in the Irish Republic after a career in the Irish army. He spent six years in the Termonbacca home run by the Sisters of Nazareth in Londonderry, from 1956 to 1962, after his parents died. He suffered sexual and physical abuse.
The delegation who met the ministers included Margaret McGukin, who was in Nazareth House on the Ormeau Road in Belfast for a decade from 1958.
"I am pleased with what went on inside (Stormont Castle). This is only one day, it is not us going away, it is just one day at a time but we are more hopeful than we were this morning," she said.
Jon McCourt, 59, who was in Termonbacca in Derry, said: "I am very optimistic about what was said today and I think they have agreed they are going to supervise the other government departments. I never thought I would see this day but I am hopeful."
SDLP Assembly member Conall McDevitt accompanied the families and said he believed the matter could be resolved.
"There is a determination in the Office of the First Minister and Deputy First Minister to begin to work on these issues," he added.
SIC: BT