Following Monday's show and the Herald's reports on the plea, about 2,000 requests for organ donor cards were made. After the interview, the writer said the rise in the take up of donor cards was "uplifting".
Mr Deasy (49) made an additional plea on Liveline on Tuesday and there was a further surge of 3,500 applicants.
Mark Murphy, chief executive of the Irish Kidney Association, said Joe Duffy did not express an opinion while highlighting the issue during the past three days, preferring to let the people who called speak for themselves.
"It worked very well because we are just inundated with calls for donor cards to a point where we have never been before. It is the power of Joe Duffy and it is wonderful," he said.
"We have never had such a level of requests at any one time. Joe Duffy is claiming it is a record. It certainly is."
On the show, Mr Duffy also listened to others in need of an organ, as well as those who have successfully undergone transplants.
The Liveline host admitted he was almost in tears listening to callers who are waiting for organ transplants.
"I've heard the most harrowing stories from people on the waiting list," he said.
"Some of the stories left me close to tears. There was one woman whose 14-year-old son has a tumour on his liver and it's getting bigger.
"I have three children the exact same age and I just think we have to do anything we can to shorten that list."
The scriptwriter was first struck down with liver cancer four years ago and he had an operation to remove the tumour; however, a scan in January showed the disease had returned.
He spoke to the Joe Duffy Show to raise awareness for the cause, which has subsequently seen a record-breaking increase of applicants to donate organs.
He said: "It is an invisible death row. The solution is there, the procedure is there, and the budgets are allocated for surgery, but the one thing they [medical teams] don't have is organs."
"They're not trying to discover a new drug, these surgeons know what to do, they just need organs."
The writer who penned Prime Suspect, The Passion and the recent hit RTE series Father and Son has a relatively rare blood group B and can receive a transplant only from this same group.
The touching words from the Scottish-based father of three prompted a staggering response from Irish listeners.
More than 600 people in Ireland are waiting for organ transplants. About 500 need a new kidney and the remainder are awaiting a heart, lung or liver transplant.
Text 50050 for a donor card