The 41-year-old was shot and killed as he got out of his car to confront a gang at the Lordship Credit Union in Co Louth, close to the border, on Friday night.
Parish priest Fr Paidraig Murphy, who has been with the family this weekend, said Garda Donohoe's widow Caroline - also a Garda - and children Amy and Niall are trying to cope.
He said: "Like anybody, how do you cope in a situation like that?
"They are gradually coming to terms with it. They have had great support from colleagues and the local community but, there is still a human element - there is a wife, Caroline who has lost her husband and children who have lost their father."
The family have described Mr Donohoe as a beloved husband and adored father.
An online death notice read: "Formerly County Cavan. Beloved husband of Caroline and adored dad of Amy and Niall. He will be forever missed by his beloved wife, daughter, son, parents Peggy and Hugh, parents-in-law Bridie and Stephen, sisters Mary and Anne, brothers Colm, Martin and Alan, brothers-in-law Derek, Kieran and John, sister-in-law Angela, nephews, nieces, relatives and friends."
A funeral service is expected to be held at St Joseph's Church, Dundalk on Wednesday.
It is like a dark cloud has descended on our parish. People are genuinely shocked and bewildered. Because he had been here for the last number of years and his children attend the local school, I think, makes it all the more poignant. He was known as a local. He was a big man with a big heart.
Fr Paidraig Murphy
Meanwhile a major, cross-border hunt for the four-man armed gang has been launched.
The PSNI have offered to help Gardaí with their investigations, with up to 100 Garda Special Detective Unit officers and the National Bureau of Criminal Investigation also involved. A cordon extending for several miles has been set up at the scene in Bellurgan, Jenkinstown.
Detectives believe a criminal gang rather than dissident republicans were responsible.
It is understood the raiders, who opened fire without warning, made off with €4,000.
Garda Adrian Donohoe was the first serving Garda to be killed in 17 years.
Churchgoers said prayers for him on Sunday morning.
Cardinal Brady, leader of the Catholic Church in Ireland said people had been shocked and horrified.
The Primate said: "I appeal to all of society to unite in rejecting such wanton violence and criminal lack of respect for human life
"The taking of innocent life is always wrong. Detective Garda Donohoe lost his life as he carried out his duty in the service of society."
Detective Garda Adrian Donohoe was escorting a cash delivery to a premises on the Cooley Road when the botched robbery took place at around 9.30pm on Friday.
He was with a colleague, who was left traumatised but unhurt after the shooting.
Garda Commissioner Martin Callinan said they are following a number of lines of inquiry.
He said: "There are up to four people involved in the actual raid. How many were involved beyond that will remain to be seen.
"We will put all our energy into finding out precisely who was behind this robbery. It is too early to speculate whether there was a subversive element to this or whether they are ordinary criminals."
Tributes have been paid by politicians north and south.
A joint statement from the Office of the First and deputy First Ministers offered "our sincere and heartfelt sympathies to his family, friends and colleagues".
Mr Robinson and Mr McGuinness added: "We must all work together on both sides of the border to make sure they are caught for this murderous deed."
Taoiseach Enda Kenny said: "(It) has left a family without a husband and father and the Garda Siochana without a brave and valued member.
On behalf of the Government and of the people I would like to express my condolences to Adrian's wife and children and to all his extended family at this unbearably sad time."