David Black, 52 and from Cookstown, is the 30th prison officer to be killed in Northern Ireland since 1974. He had served for more than 30 years and was nearing retirement.
On Thursday evening, his widow Yvonne and their son and daughter said "sadness and grief in another home will achieve nothing".
Mr Black is understood to have been travelling to work at Maghaberry high-security prison - where protests by dissident republican inmates have been ongoing - at about 7.30am on Thursday.
He was driving a black Audi A4 (registration HHZ 1560), when a dark-blue Toyota Camry with a Dublin registration (94 D 50997) pulled up alongside and a number of shots were fired.
The victim's car immediately veered off the road and into a deep drainage ditch.
Police have examined the scene and are also investigating a car - believed to be the Dublin reg Camry - burnt-out in the Inglewood area of Lurgan, Co Armagh.
At this stage, officers believe Mr Black sustained "very serious and fatal gunshot wounds" in the attack. A post-mortem is due to be carried out.
This morning, a prison officer - David Black - whose profession is simply to keep people safe, was brutally murdered on his way to work.
Chief Constable Matt Baggott
"Please be assured that the Police Service of Northern Ireland will do everything possible to bring those responsible to justice," Chief Constable Matt Baggott said.
"I would ask everybody to get right behind the murder investigation as the hours and days unfold. This is an attack on all people and we need the full support of everyone."
Assistant Chief Constable Drew Harris added: "The motive behind this is sheer terror."
Reverend Tom Greer, of Molesworth Presbyterian Church, told UTV that Mr Black "was a lovely guy to know in so many ways.
"David was not a man to make enemies of anyone.
"He was a guy whose friendship I and others in our church valued greatly."
The father-of-two's murder has been condemned by the First and deputy First Ministers, by representatives of parties across political divides and by the Dublin and London governments.
Justice Minister David Ford branded the murder "horrific".
"At the outset, I want to express my deepest sympathy to the family and colleagues of the murdered prison officer," he said.
"This is a horrific murder, for which there can be no justification, and it will be rightly condemned by all across Northern Ireland."
Mr Ford added: "At its very core, this is a human tragedy. A man has lost his life this morning and a family are grieving the loss of their loved one. My thoughts are also with the wider Prison Service family.
"We need to respond to this horrific act by bringing those responsible to justice."
Kate Carroll, the widow of PSNI Constable Stephen Carroll who was shot dead by dissidents in 2009, told UTV that she had feared another attack against a member of security forces would happen.
"I can't even imagine how those people are going to feel but, from my perspective, it was the worst news I had ever been given in my life," she added.
You knew that it may happen, just because of the wearing of the uniform ... When you see the people going to deliver that news, it's devastating.
Chair of the Policing Board Brian Rea said members were unanimous in their condemnation of the prison officer's killing.
"This man's family, friends and colleagues have received the worst news possible and our thoughts are with them all at this terrible time," he said.
"There can be no justification for this murder. As a community we have travelled too far to allow the progress made to be undermined by ruthless people.
"That is the overwhelming view of the majority."
Mr Black was a member of the Orange Order and the institution says his death brings the number of members murdered in terrorist attacks since 1969 to 337.
Grand Master of the Grand Orange Lodge of Ireland, Edward Stevenson, paid tribute to the victim.
"David was a devoted and loyal member of Montober LOL 661 for approximately 30 years, as well as a fine prison officer," he said.
"His professionalism throughout the worst of the Troubles, and beyond, is in stark contrast to the cowardly and faceless terrorists who today have left a wife without her husband and two children without their father."
The M1 was closed in both directions for much of the day, but has since reopened.
Police are urging anyone with information to call 0845 600 8000 and ask for the murder incident room, or to pass information anonymously through Crimestoppers on 0800 555 111.