On February 3, officers were briefed a safety alert had been issued for Allan and he was likely to be in possession of firearms.
Officers were told they could carry firearms if necessary as a result.
In his findings, authority chairman Judge Colin Doherty said the officer was aware a safety alert had been issued for Allan and he was considered a "significant risk" to police officers and the public.
"Given Mr Allan's decision to present a pistol at the officers in a manner indicating an intention to shoot, it was not practical or realistic for the officer to use a lower level of force."
Six of the shots hit the bonnet of the vehicle Murray was in, while the other hit the windscreen, and a fragment of the bullet hit Murray in the side of his face.
Canterbury district commander Superintendent John Price said they accepted the findings and supported the actions of the officers.
"The officer followed correct police processes and established that Mr Allan posed an immediate threat of death or grievous bodily harm and that he needed to use his firearm to defend himself and his fellow officers," he said.
"The officers involved in this situation not only ensured their own safety, but the safety of their colleagues and the wider public, in what was an extremely high pressure situation.
"I fully support their actions and am proud of the way they conducted themselves during this incident."
The authority also found the officers were justified in arming themselves at the start of their shift, and said Allan received the appropriate medical assistance after he was injured.
In August, Murray was jailed for 18 months after pleading guilty to assault with a weapon, unlawful possession of an imitation firearm and ammunition, as well as three charges of dishonestly using stolen cheques.
In June, Star.kiwi revealed Murray was a convicted sex offender.
Records show in January 2013, the then 19-year-old was jailed for three years and two months for having non-consensual sex with a 14-year-old in what was described as "very close to rape".
The court was told the girl was emotionally traumatised, and had tried to commit suicide twice following it.
At the time, Allan's pre-sentence report said he could cause a "high level of harm" if he offended again.
Allan was released on parole on December 3, 2015, with conditions he would not have any unsupervised contact with under-16s, or with the victim.
- Star.kiwi