The Primate of All Ireland told UTV the priest was challenged about his alleged involvement in the Claudy bomb.
Fr Chesney, who died from cancer, aged 46, in 1980, was never questioned by police despite being suspected of masterminding the atrocity.
Police at the time were reluctant to arrest the cleric for fear of inflaming the security situation in 1972.
Fr Chesney was called in for questioning by the-then Bishop of Derry Neil Farren and his successor, Edward Daly, and denied any involvement in the attack.
He was transferred to a parish in Co Donegal, outside the Northern Ireland jurisdiction in 1973, following secret talks between then Secretary of State William Whitelaw and the head of the Catholic Church in Ireland, Cardinal William Conway.
The two men discussed the case after being approached by a senior RUC officer.
"I do not think the Catholic Church was involved in a cover-up. It received this request from the highest civil authority here to help in a very difficult situation", Cardinal Brady told UTV on Tuesday.
"He (Cardinal Conway) had him questioned, not once but twice, which is more than the police did, and brought back his findings to the Secretary of State and that is where the investigation ceases. Now what more Cardinal Conway could have done is not very clear to me," he said.
On Tuesday, a Police Ombudsman report found that the RUC investigation into Fr Chesney was "compromised" after senior officers conspired with the Government and Church to protect him.
When asked by UTV if Fr James Chesney got away with murder because of the failings of the State, the police and the Catholic Church, Cardinal Brady replied:
"That allegation has been made, but as we see, it hasn't unfortunately been investigated and that is a terrible pity".
Bishop of Derry Séamus Hegarty said he was "shocked, ashamed and annoyed" that a priest would have been in any way associated with the atrocity.
Bishop Hegarty said: "For a priest to be involved or to have been alleged to be a part of it is something that runs totally counter to everything that we stand for.
"It is a great sadness and a great sorrow to all of us."
"It was widely known that Father Chesney had certainly republican tendencies," Bishop Hegarty also told UTV.
"But the real question to be asked is: how is it that the police who had a lot of intelligence and who were well-informed of what was going on, how is it that they did not arrest him and question him?
"It was not for the Church to pry into what was a criminal matter. We did not have the competence to do so."
In a joint statement, Cardinal Brady and Bishop Hegarty said the action of the Church "did not prevent the possibility of future arrest and questioning of Fr Chesney".
"In the course of this enquiry the Police Ombudsman's investigation found no evidence of any criminal intent on the part of any Church official", they added.
On Tuesday, Ombudsman Al Hutchinson revealed that the police had intelligence suggesting that Fr Chesney continued to be involved in the IRA after the Claudy bombing.
He did not elaborate on details and noted that the period was not within the remit of his investigation.
In July 1972, the no-warning atrocity in the quiet Co Londonderry rural village claimed the lives of nine people and injured 30 more - Protestants and Catholics, young and old.
Catholic families affected by the bombings did not attend a press conference with other relatives and survivors on Tuesday, after the briefing by the Ombudsman's team.
But one Protestant survivor said the Catholic community in Claudy had been "doubly betrayed" by the alleged involvement of a priest in the atrocity, as calls were made for a fresh investigation.
Ulster Unionist councillor Mary Hamilton - who, along with her husband, owned the Beaufort Hotel which was destroyed by one of the blasts - said the Ombudsman's report was only the beginning.
"The Claudy community has suffered greatly, but has remained united. I would like to express my heartfelt sympathies particularly to those within the Catholic community who, having endured this attack, were doubly betrayed by the discovery that their own priest was involved," she said.
"The perpetrators were able to live out their lives until God took them - and could be walking among us still - but that is much more than the victims were allowed."
She added: "This has been an emotional day, but we have as many questions as ever.
"I will continue to press for those to be answered - the lives of the people of Claudy are no less valuable than those lost during Bloody Sunday."
Mark Eakin, who was injured in the blast that killed his younger sister Kathryn - the youngest victim of the atrocity - said he also felt for the Catholics who had to hear the shocking revelations about a member of their faith.
"Claudy was a mix and match, there were five Catholics killed and only four Protestants," he said.
"It was a bad day for everybody and I just feel so sorry for some of the Catholic people that had to sit up there today and listen to what they had to listen to about their own Church. I feel they've been let down by their Church."
He added: "I feel I have been let down by the Government that I pay my taxes to.
"They have not performed at all; they have totally washed their hands of Claudy and preferred to wash it under a carpet for 38 years."
Marjorie Leslie, who was injured during the explosions, backed calls for a fresh investigation.
Speaking about Fr Chesney's role in the bombings, she said: "He didn't do it alone."
No-one has ever been charged with the Claudy murders.
From a nine-year-old girl to a 65-year-old street cleaner, the triple car bomb attack in what was a quiet rural village claimed the lives of nine people - Protestants and Catholics, young and old.
Kathryn Eakin, 9, Protestant. The youngest of the victims, she died in the blast from the first bomb on Main Street. She had been cleaning the windows of the local grocery shop which was owned by her family.
Patrick Connolly, 15, Catholic. The teenager was caught up in the first explosion outside McElhinney's pub and shop. He died in hospital just over a week later.
William Temple, 16, Protestant. From nearby Donemana, the boy worked as a milkman's helper and was killed while on his round in the village.
Arthur Hone, 38, Catholic. A married father-of-two and an insurance salesman, he died a fortnight after the bomb attack. Two of his uncles - both priests - conducted a requiem mass at his funeral.
Joseph McCloskey, 39, Catholic. A factory worker, he died instantly when the first bomb went off.
Rose McLaughlin, 52, Catholic. A mother- of-eight, the café owner died in hospital four days after the outrage.
Elizabeth McElhinney, 59, Catholic. The owner of the pub and shop where the first bomb was planted, she was serving petrol from the shop's pump when she was killed.
David Miller, 60, Protestant. A street cleaner, he was killed by the third and final bomb contained in a mini-van.
James McClelland, 65, Protestant. A street cleaner and the oldest of the victims, he died in the third explosion.
At the Claudy bomb inquest, a coroner described the outrage as "sheer, unadulterated, cold, calculated, fiendish murder".