The cleric, whose final years at the head of the clergy were dogged by abuse scandals, handed in his resignation on age grounds last month after turning 75, as required under canon law.
The rapid acceptance of a bishop's resignation within one month is unusual for the Catholic Church but his departure has been flagged for some considerable time.
In the last few years, Cardinal Brady faced repeated calls from clerical sex abuse survivors to quit over his involvement in the Brendan Smyth case.
The teacher and canon lawyer swore two victims of the notorious paedophile priest to secrecy during an internal church inquiry in 1975 into the abuse of two children.
Their evidence was never handed over to police, allowing Smyth to continue abusing other youngsters countless times before he was finally jailed in 1994.
The Cardinal insisted his role in the canonical inquiry was as a notetaker.
In a farewell message at mass in St Patrick's Cathedral in Armagh the church leader said he recalled Pope Francis' motto "miserando atque eligendo" which he said "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," Cardinal Brady said.
"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."
Last month Pope Francis promised to hold bishops to account for the protection of children and begged forgiveness from victims after he celebrated a Mass with six survivors at the Vatican.
One of them, Irish woman Marie Kane, said she asked the pontiff to remove Cardinal Brady from his post because of the way he handled abuse allegations.
Cardinal Brady led the church in Ireland for more than 17 years during which time a series of investigations exposed shocking levels of clerical abuse.
Archbishop Eamon Martin will take over as Archbishop of Armagh and Primate of All Ireland - the 116th man to fill the role.
Archbishop Martin, is a 52-year-old Derry-born former teacher who was named as Coadjutor Archbishop of Armagh early in 2013.
Three years earlier, at the height of pressure on Cardinal Brady over his handling of the Smyth case, the church leader asked Pope Benedict to ease his workload by appointing someone to a support role.
Archbishop Martin was only formally put in place as the assistant 16 months ago.
Last December an investigation by church watchdog the National Board for Safeguarding Children said Cardinal Brady made a ''commendable decision to gather and document whatever information was available" about abuse allegations in his own archdiocese on taking up his role as Primate of All Ireland in 1996.
At the time he said he was truly sorry for the suffering of victims.
Although he has also apologised to Smyth's victims, he previously said he would not resign over the affair.
Archbishop Martin paid tribute to the Cardinal and described him as a gentle and humble man who is never fully comfortable in the limelight.
"This is not just my day. It is a day for us to recognise the years of service which you have given to the Church in Armagh and beyond," Archbishop Martin said.
"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."
Archbishop Martin said his appointment was an honour.
As a priest he taught at St Columb's College, Derry and went on to become administrator of the Diocese of Derry in November 2011. He has also served as a member of the Catholic Bishops' Joint Bioethics Committee and a director of the National Board for Safeguarding Children in the Catholic Church in Ireland.
Archbishop Martin has a keen interest in music, particularly in choral and classical as well as liturgical and sacred, especially Gregorian Chant. He also enjoys walking and gardening.
The Most Revd Dr Richard Clarke, Church of Ireland Archbishop of Armagh and Primate of All Ireland, praised Cardinal Brady's ecumenical work.
"He has been a good friend to successive Archbishops of Armagh and to the wider Church of Ireland throughout his archiepiscopate, and we are grateful to him for this unaffected generosity of spirit," he said.
"And, on a personal note, I wish to thank Sean for real kindness and warm friendship over many years. We all hope that he will enjoy both true fulfilment and good health in the years ahead."
Archbishop Clarke said he looks forward to working with his new Catholic counterpart in Armagh.
Presbyterian Moderator in Ireland Right Rev Dr Michael Barry said Cardinal Brady will be missed.
"Dr Brady was committed to building good relationships with the Protestant Churches and their leaders. On behalf of the Presbyterian Church in Ireland, I would like to wish him a long, peaceful retirement and that he would know God's blessing in the days ahead," he said.
Rev Dr Donald Watts, co-chair of the Irish Inter Church Meeting and president of the Irish Council of Churches, said: "I would also wish to acknowledge his enormous contribution to inter church relations in Ireland.
"He chaired meetings with a gentle humility that has enabled rich dialogue and built understanding. As he retires we enjoy unprecedented positive relationships between our member churches."Suggest a correction