Dr Brady joined prelates from Italy, Germany, the US, Spain, India, Argentina, Kenya, Mexico, Poland, Senegal, Brazil and France.
During the Consistory ceremony in Saint Peter's Basilica, the new cardinals, known as the 'princes of the Church', took it in turns to kneel before Pope Benedict XVI and receive the cardinal's scarlet hat, or berretta.
While cardinals will maintain their existing roles, they will also gain substantially more important responsibilities as those closest to the Pope within all the ranks of the Catholic Church.
The Anglican Archbishop of Armagh and Primate of All Ireland, the Most Rev Alan Harper, welcomed the elevation of Dr Brady to Cardinal, saying it had given him "the very greatest joy and satisfaction".
"We in Armagh know well the quality of the Archbishop's ministry and the respect and affection in which he is held throughout the community," said Archbishop Harper, praising him as a "colleague of generosity, warmth and true spiritual depth".
During Saturday's ceremony, Pope Benedict pressed particularly for an end to the war in Iraq and spoke up on the sufferings of the country's Christian minority.
The Baghdad-based Patriarch of Babylon of the Chaldeans, Emmanuel III Delly, was one of the prelates to be appointed Cardinal.
Since the start of the war, large numbers of Iraqi Chaldeans - the largest Christian group in Iraq - have fled the country and the Vatican remains concerned that they could disappear altogether as a result of the hardship being created by the violence.
Pope Benedict, in gold embroidered vestments, said during his homily: "Let us together reaffirm the solidarity of the whole Church with the Christians of that beloved land and invoke from the merciful God the coming of longed-for reconciliation and peace for all the peoples involved (in the conflict)."
