The statement comes after the publication of a report by Sir Anthony Hart of the Historical Institutional Abuse Inquiry.
The report concluded that victims of historic child abuse in state, church and charity run homes between 1922 to 1995 should receive state-backed compensation payments of up to £100,000.
Inquiry chair Sir Hart said the minimum pay-out should be £7,500 with the maximum amount given to those who had experienced severe levels of abuse.
Part of the investigation focused on the activities of paedophile priest Brendan Smyth.
During evidence sessions the inquiry heard lurid details about the activities of the serial child molester who frequented Catholic residential homes and was convicted of more than 100 child abuse charges.
Sir Anthony said despite knowing his history of abusing children, the Norbertine religious order only action was to move Smyth to different diocese where he then abused more children.
They failed to report the abuse to police "enabling him to continue his abuse", the report found. The Order also failed to take steps to expel him from priesthood.
Archbishop Eamon Martin, Primate of All Ireland said that the report was a "significant moment" for survivors and their families.
Archbishop Martin commended the survivors for the courage and dignity they displayed throughout the inquiry process.
He went on to apologise to the survivors.
The statement reads: "I apologise unreservedly to all those who suffered from their experience in Church-run institutions, and to their loved ones.
"I know well that my words are inadequate in attempting to address the enormity of the harshness and brutality which many innocent children experienced. There is never an excuse for the abuse and ill treatment of children or any vulnerable person, in any setting. When the perpetrator is a priest or religious, it is also an appalling betrayal of a sacred trust.
"I am ashamed and I am truly sorry that such abuse occurred, and that in many cases children and young people felt deprived of love and were left with a deep and lasting suffering."
The Archbishop added that the Church must do everything they can to demonstrate they are serious about making reparation to the "sins and crimes of the past".
Bishop Leo O'Reilly of Kilmore condemned the crimes committed by abusers as "appalling".
He added: " As a minimum we owe to those who lost their childhood through abuse the highest of safeguarding standards along with a commitment to assist them in every way possible to find healing and freedom from the abuse and pain so cruelly inflicted on them in their childhood.
"On behalf of the diocese of Kilmore I am committed to engaging with other bodies in implementing the recommendations in the report to the best of our ability."