"In hindsight we now know he probably never would have recovered from those injuries, they were terminal," Lieutenant-General Jones said.
The New Zealand Defence Force had given an undertaking to Corporal Grant's family that he would not be named until today.
He leaves a wife and two young children - a 7-year-old daughter and a 5-year-old son - and his death is the first SAS fatality in Afghanistan in four deployments since 2001.
Corporal Grant's widow Tina is also in the Defence Force in an educational capacity and the pair are understood to have met while she was a medic serving alongside the SAS.
In a statement, Corporal Grant's family acknowledged and expressed appreciation for the support they had received from the NZ Army.
"Doug had a wide group of friends from all walks of life. The support and love from all of these people has been humbling."
They said Corporal Grant "had absolute faith in his friends and colleagues, and what he was doing in Afghanistan".
"He understood what he was getting into and believed in the goal of training local forces for that country's future.
"He was a determined person, some may say stubborn in his pursuit of his goals while growing up. He wrote in a school essay once that he wanted to be in the SAS; he worked towards that goal from that time onwards and convinced the Army he could do the job.
"The family is incredibly proud of Doug's achievements as a soldier and as a husband, father, brother, and son."
Corporal Grant was determined to keep his work life separate from his family life, and his family asked that his widow and children be given "some private space to come to terms with the fact he has died doing one of the things he loved".
Corporal Grant was a long-serving member of the SAS, with a total of 21 years with the Defence Force a third of that with the SAS. He was on his 2nd deployment with the unit in Afghanistan.
He'd also served with the Royal NZ Infantry, Royal NZ Engineers and SAS, as well as two tours to East Timor and one in former Yugoslavia.
A "competent and likeable'' soldier, Corporal Grant had left the SAS some time ago out of concern for his young family but had returned, "because that's how he saw himself - as being an SAS soldier from the start and the finish", Lieutenant-General Jones said.
Corporal Grant's SAS colleagues were grieving but retained their enormous self confidence and self discipline.
"The work is always dangerous and they're brave men who do their best to serve New Zealand."
Corporal Grant's body is returning home via commercial flights.
Details of his return and funeral arrangements are yet to be confirmed but they are likely to include a private commemoration in Auckland before a more formal ceremony at Linton army base where he will lie in state for several days.
Today Lieutenant-General Jones said: "The events of Friday in Kabul reinforce to us all that ours is a dangerous profession, whether on deployment or in training and while we accept these risks the death of a colleague, friend and father is always difficult to take.
"Our thoughts are with his family, members of his unit and friends and colleagues within the New Zealand Army and New Zealand Defence Force as a whole."
Prime Minister John Key, Defence Minister Mapp, chief of Defence Force Lieutenant-General Rhys Jones and SAS Commanding Officer Lieutenant-Colonel Chris Parsons visited his family on Saturday.
Mr Key said yesterday that Corporal Grant's wife had relayed to him "his absolute passion for what he was doing and the belief in the task he was undertaking in Afghanistan".
"I don't know if any of that makes it easier. At the end of the day a family has lost their father and we are obviously deeply saddened by that."
Nine people are reported to have been killed and 22 injured in the attack on the British Council headquarters in Kabul, which began when car bombs opened up the wall surrounding the building, allowing Taleban terrorists to rush in with weapons firing.
The Guardian reported Taleban spokesman Zabiullah Mujahid as saying it was a symbolic act timed to coincide with the annual celebration of the day in 1919 when Afghanistan won the right to run a foreign policy independent of Britain.
"We attacked the buildings because we want to remind the British that we won our independence from them before and we will do it again."
The SAS had been due to end its deployment this March but it was extended a year, allowing more time for the SAS to mentor the ready response unit of the Afghan police, the Crisis Response Unit.
They have been involved in a suicide raid in June on the Intercontinental Hotel and a raid last month on the home of Jan Mohammad Khan, an adviser to President Harmid Karzai.
Mr Key said yesterday the Taleban were clearly changing their tactics, and were undertaking more high-profile operations.
He also said that after speaking to Lieutenant-Colonel Parsons yesterday, the SAS were increasingly impressed with the way the Afghanistan Crisis Response Unit (CRU) was operating.
"Their level of performance actually at the British Council was very, very high on Friday."
Mr Key said SAS feedback showed there had been a dramatic increase in performance from the CRU since the attack at the Intercontinental Hotel.
"A number of countries" had indicated they would take over the work of the SAS mentoring the CRU when it left "so I don't think it is likely they would be on their own".
