Irishman Sir Terry died on Sunday after a "short but brave battle with cancer".
Born in County Limerick in 1938, he had a career in radio and television that spanned 50 years, and included his BBC Radio 2 Wake Up To Wogan programme to which Fr D'Arcy contributed.
Fr D'Arcy said Sir Terry "perfected the art of radio from a very early age".
"He was the only man I ever met who could make a pause sound interesting," the County Fermanagh priest told BBC Radio Ulster's The Sunday News programme.
"Terry actually milked it and everybody was hanging on his every word after that.
"He was just that kind of broadcaster who made everybody feel at home and feel as if they were the only one in the world he was talking to."
An emotional Fr D'Arcy visited Sir Terry on Thursday "to say my goodbyes".
He said it was "a sad but beautiful occasion".
"The family will be awfully, awfully sad because the centre of his life was his family, his children and his wife Lady Helen especially."
As a young fan and before he was ordained, Fr D'Arcy travelled to Rathmines in County Dublin to see Sir Terry's marriage to Helen in 1965.
The two men became friends a short time later, but as Fr D'Arcy said, they often joked about how long their friendship had lasted.
"The two of us could never remember, we used to talk about that.
"[Terry] said: 'What's the difference between 40 and 50 years? We're two old curmudgeons lucky to be alive.'"
Sir Terry started his broadcasting career with the Irish state broadcaster RTÉ, where he became friends with Irish icons Gay Byrne and Larry Gogan.
"He began in a very dull, old building in Henry Street in Dublin, a terrible place where mice would appear in the middle of his broadcast," Fr D'Arcy said.
"One day the great Mike Murphy was reading the news and Terry went in and set the bottom of the news sheet he was reading on fire.
"We didn't think his broadcasting would work in England but he actually loved England, loved broadcasting and nothing better than getting reaction from his people.
"As he used to say, the scriptwriters for his programme were the listeners - they sent in hundreds [of faxes and emails]."
And according to Fr D'Arcy, tales of Sir Terry's antics will live long in the memories of his friends and many fans.
"There will be Terry Wogan stories for eternity," he said.
"It was incredible that he would take a morning programme and, by codology and foolology, lighten the days of over eight million people.
"That's a legacy that anybody would be proud of."
Gay Byrne, best known for presenting television and radio programmes in the Republic of Ireland, said Sir Terry was "born with a monster advantage over the rest of us".
"He had a permanently funny disposition - he saw the fun in everything, and this is a huge gift to have," he told RTÉ.
"He probably was the most popular and most listened-to broadcaster in the world - he had a huge listenership right across Europe and he had more listeners than all the rest of us combined."
A large part of that was down to his unique voice, Gay said, that had been honed by reading radio news bulletins early in his career.
"He had a lovely voice and it produced a lovely, round, mellow sound.
"People found it attractive to listen to."
Television presenter Gloria Hunniford, from Northern Ireland, also worked with Sir Terry through the decades.
The pair developed a close friendship during her spell on Wake Up To Wogan, and she said his death felt like "losing a member of my family because I seem to have known Terry forever".
"I have to be very grateful to Terry for all sorts of things apart from his friendship, because he turned my broadcasting career around," Gloria said.
"He was very generous of spirit, a terrific family man and had a unique sense of everything."
Sir Terry moved to England in the late 1960s to begin work with the BBC, but always kept Ireland close to his heart, Gloria said.
"Terry always talked about his Irish roots, he loved anything to do with an Irish story and he loved Irish music," she said.
"There'll never be anybody like him in broadcasting again, I believe."
Irish singer Daniel O'Donnell said he and other artists from Ireland owed much of their success to Sir Terry as he heralded them on his programmes and brought their music to his huge UK audiences.
"Back in the late '80s, I was on his TV show a couple of times, and it was a great help to me because I was just starting out," he said.
"He was very important to a lot of the singers and acts, and I'm sure he had a great part to play in our being on [British] shores.
"He never forgot where he came from even though a lot of people just saw him as a presenter in the UK.
"Terry was very aware of his roots and where his life began, and for that you have to admire him greatly."
Paul O'Grady, the BBC Radio 2 presenter and chat show host, also originates from the Republic of Ireland, like Sir Terry.
He described him as "a wise old bird" and said their conversations "usually went to Ireland".
"He was a brilliant broadcaster, whether it was on the screen or on the radio," Paul said.
"The man could do it - he was the master, he truly was."
Sir Terry Wogan "was the one who made radio in an age of television", his close friend and fellow broadcaster Father Brian D'Arcy has said.