Victim support groups have welcomed the decision after the 75-year-old sought retirement at the age of 75 last month.
He had been heavily criticized for his part in the Fr Brendan Smyth case when he swore two victims of the pedophile priest to secrecy during an internal church inquiry in 1975.
State broadcaster RTE reports that their evidence was never handed over to police. Smyth continued to abuse children until he was finally jailed in 1994.
Cardinal Brady has insisted his role in the canonical inquiry was as a notetaker, but victim groups have campaigned against the priest, who has held the role of Archbishop of Armagh since 1996.
RTE reports that Cardinal Brady used his farewell message at mass in St Patrick’s Cathedral in Armagh to say sorry.
He said: “I recall Pope Francis’ motto ‘miserando atque eligendo’ which challenges and inspires me with its message of God having mercy and at the same time choosing us, despite our sinfulness.
“It reminds me that I too need to say sorry and to ask forgiveness. And I do so again, now.
“At the same time, Pope Francis’ motto inspires me to trust in the mercy of God and to pray for the strength to do always as Jesus would have me do.”
Archbishop Eamon Martin, a 52-year-old from Derry, will take over as Archbishop of Armagh and Primate of All Ireland, the 116th man to fill the role.
He said: “This is not just my day. It is a day for us to recognize the years of service which you have given to the Church in Armagh and beyond.
“On behalf of the people, priests and religious of the Archdiocese of Armagh, I want to thank you sincerely for serving us with love and dedication.
“We appreciate all that you have done for us and we assure you of our continued affection and prayers. We wish you every blessing for a healthy and peaceful retirement.”
Marie Kane, a survivor of clerical child sex abuse who told Pope Francis that Cardinal Brady should stand down, has welcomed the Pope’s acceptance of the Cardinal’s resignation.
In a message to RTÉ News, Kane said: “Good to hear Seán Brady’s letter of resignation has been accepted.”