He also highlighted that no state or church guidelines existed in the 1970s in the Irish Republic to assist those responding to an allegation of abuse against a minor.
The BBC documentary reported the testimony of Brendan Boland, a 14-year-old victim of Father Smyth, arrested in 1994 and convicted in 1997 of sexually assaulting 20 victims over a period of 35 years.
In 1975, Boland told a three-priest inquiry team, which included the then-Father Brady, of his two years of abuse at the hands of Father Smyth. This became public knowledge in 2010 and led to calls for the cardinal to step down over the oath of secrecy that Boland was forced to sign and the fact that the civil authorities were not informed of the abuse.
The BBC program reported that, during his deposition, Boland also furnished the inquiry with the names and addresses of other victims of Father Smyth.
According to journalist Darragh MacIntyre's report, the parents of these victims were never notified by the church of the abuse allegations.
One of the victims said in the documentary that he was sexually abused for a further year by Father Smyth after the inquiry was completed, while his sister was abused until 1982 and that four of his cousins were abused until 1988.
According to Cardinal Brady, when the inquiry was completed he passed all the information he had obtained to his bishop, Bishop Francis McKiernan.
In his May 2 statement, Cardinal Brady rejected the program's claim that he was an investigator in the inquiry.
"I did not formulate the questions asked in the inquiry process. I did not put these questions to Mr. Boland. I simply recorded the answers that he gave," he said.
"The documentation of the interview with Brendan Boland, signed in his presence, clearly identifies me as the 'notary' or 'note taker.' Any suggestion that I was other than a 'notary' in the process of recording evidence from Mr. Boland is false and misleading," Cardinal Brady said.
He said he subsequently interviewed one of the alleged victims who lived in his own diocese.
"That I conducted this interview on my own is already on the public record. This provided prompt corroboration of the evidence given by Mr. Boland," he said.
The cardinal also said it was incorrect to suggest that he had the "power to stop Brendan Smyth in 1975."
He said even Bishop McKiernan had limited authority over Father Smyth, and that those culpable for the inadequate response were the Norbertine abbot and religious superiors.
Describing himself as "shocked, appalled and outraged" when he "first discovered in the mid-1990s that Brendan Smyth had gone on to abuse others," he said he thought that Bishop McKiernan had taken the evidence to the abbot of Kilnacrott and that the abbot would then have prevented Father Smyth from abusing others.
In an interview with RTE Radio May 2, Msgr. Charles Scicluna, the promoter of justice in the Congregation for the Doctrine of the Faith, defended Cardinal Brady's handling of the case.
"He was doing his duty to investigate something that had come to the knowledge of the church, and I think he fulfilled his duty well," he said, adding that the then-36-year-old priest acted as a notary, not an investigator.
Cardinal Brady did say that he was part of "an unhelpful culture of deference and silence in society and the church" which he said was now "a thing of the past."
In the wake of the publication of the Murphy Report in 2009, Cardinal Brady told RTE that he would resign if he found that a child had been abused as a result of any managerial failure on his part.
Two years ago, the cardinal refused to resign, offering to remain on to lead the church in Ireland forward on its path of renewal as a "wounded healer." He suggested he was better placed to help it deal with the tragedy of child sexual abuse on account of his brokenness.
Marie Collins, an abuse survivor from Ireland who recently participated in a Vatican symposium on abuse, said she was repeating her 2010 call for Cardinal Brady to resign.
"What I saw in that documentary was just appalling. He has to go and, if he doesn't, how can a man like that lead the church in Ireland?" she said.
"I was devastated by Msgr. Scicluna's comments in which he backed Cardinal Brady. I was so impressed with him at the symposium on abuse in Rome in February, and then to hear him defend the indefensible. They are circling the wagons, and the Vatican has decided that Brady cannot go because they are afraid of the domino effect.
"The church in Ireland has no credibility left," she said.