The footage was provided to animal welfare group Paw Justice a fortnight ago.
The council said the dog was not mistreated and that its policy on dog euthanasia is that if an impounded dog is not claimed within seven days, and is deemed by an officer as unsuitable for re-homing, the dog will be euthanised by "humane" means.
However Paw Justice's co-founder Craig Dunn said he was disgusted by the video.
"It's a dog being euthanised with a captive bolt gun," he said.
"I've asked experts why it's been done and they can't give a medical reason."
Dunn said he believed captive bolt guns should not be used to put down dogs.
"When we're dealing with companion animals, the ideal way for myself and I think for many New Zealanders will be that the animal is taken to a vet, given a sedation pill and then it's euthanised by injection.
"The captive bolt gun is brutal. It can go wrong."
Dunn called for a nationwide policy on how local councils euthanise animals.
Council chief executive Richard King rejected the criticism and said the captive bolt mechanism was a "humane" euthanasia method that was safer for staff and for dogs.
"The dogs being euthanised are often dangerous, which means transportation from the pound facility to a vet could put staff at risk of being attacked. Further, these dogs may have been mistreated, or have never been to a vet before - the trip to the vet is often stressful for them.
"This means the dog needs to be sedated upon arrival, for the safety of the dog as well as for the veterinarian and council staff. The lethal injection then takes longer to render the dog deceased."
King said the council policy was to arrange to rehome dogs of a good temperament. Generally only dogs that were believed to be "menacing and dangerous" were destroyed.
"In the past, council euthanised dogs by using firearms or by taking the dogs to a vet to receive a lethal injection. However, the process of lethal injection was stressful for both the dogs and staff."
Animal behaviour expert, Dr Elsa Flint, said she believed using a bolt gun could not be considered a humane way to put down an animal.
"Using captive bolt as a method of euthanasia in dogs is unacceptable. It's not designed for that and should never be used. The most humane way of euthanising a dog is by lethal injection."
Flint also referred to the World Society for the Protection of Animals, which stated captive bolt guns were not acceptable nor were they recommended for euthanising dogs and cats, as other methods were more humane.