Adrian Ismay, 52, was seriously hurt after a booby-trap device exploded under his van in Belfast on 4 March.
The police said that a post-mortem examination showed he died as a "direct result of the injuries" of the bomb.
A dissident republican group, widely referred to as the new IRA, said it carried out the attack.
The organisation said Mr Ismay was targeted for training officers at Maghaberry Prison near Lisburn, County Antrim.
Sue McAllister, director general of the Northern Ireland Prison Service, spent time with the Ismay family on Wednesday.
She said they were "devastated and beyond shock at what's happened".
"Like all of us, they thought that he was on the mend and was doing really well," said Mrs McAllister.
"We thought he would make a full recovery, we expected him to come back to work in due course, so we're all shocked, but particularly Sharon and her daughters."
Mrs McAllister condemned the behaviour of some paramilitary inmates at Maghaberry Prison, who reportedly cheered, smoked cigars and mocked wardens at the high-security jail when news of Mr Ismay's death filtered through.
"This was just unacceptable behaviour by a very small number of prisoners," said Mrs McAllister.
"My colleagues in Maghaberry dealt with it professionally, with great fortitude and with the courage they display on a daily basis."
She admitted she and her colleagues in the Prison Service felt "very angry" on hearing the reports of the celebrations in the exercise yard of the prison.
"We encounter challenging behaviour by a small number of prisoners every day, and we deal with it with great professionalism and we do not stoop to behaving that way in response," she added.
A man appeared in court on Saturday charged in connection with the bomb attack on Mr Ismay.
Det Ch Insp Richard Campbell said it was "yet another difficult day for the Ismay family, his friends and colleagues as they struggle to come to terms with the events of the past 12 days".
He added: "We have spoken to the family this morning and advised them of this development.
"We are also liaising with the Public Prosecution Service in relation to the individual who is currently charged with attempted murder and causing an explosion with intent to endanger life."
The PSNI have also repeated an appeal for information on two cars that police believe may have been used by the bombers - a red Citroen C3 and a silver Skoda Fabia.
The PSNI has launched a murder inquiry into the death of a prison officer who died 11 days after being injured in a dissident republican bomb attack.