She launched it on Sunday night and it will be presented to council in two months' time.
Ms Goldsworthy said she was a dog lover with two dogs of her own.
"I always knew that gassing was carried out at the pound but I find that horrendous," she said.
"Dogs are often frightened and distressed while being crammed into the gas chamber, usually in multiple amounts. What is humane about this?" she said.
"The Wanganui District Council needs to stand up, take notice and make the necessary changes to provide the pound with the resources they need so these animals, where necessary, can be euthanased humanely by injection," Ms Goldsworthy said.
Jonathan Barrett, council's deputy strategy and development manager, said the dogs were euthanased in a special chamber, using carbon monoxide gas which sends the dogs to sleep and was a method verified as humane by Massey University vets and SPCA head office.
"Wanganui District Council euthanases a large number of unwanted dogs or dogs unsuitable for rehoming.," Mr Barrett said.
"We don't use firearms because this method is stressful to both staff and animals and may not be immediately effective. And the injection method of euthanasing requires qualified veterinary personnel," he said.
The petition has the support of Wanganui district councillor Jack Bullock, who said he would "champion" the petition when it went to council.
Mr Bullock said the pound put down an average of 25 to 35 dogs each month and the method used at the city pound was out of touch with other centres.
"I can't believe Massey University or the SPCA find this an acceptable method. It's slow murder in my opinion," he said.
"The council says it's a humane way to put dogs down after a Massey University person approved it, but I can't agree with that at all. I've witnessed this at the pound and it's far from humane.
"Imagine bundling a dog into a metal cage, closing the doors up so it's dark, then cranking up an old motor next to start the gas process, which takes 10-20 minutes. And sometimes after all that time the dog isn't dead.
"That's not humane. It's a slow, lethal and cruel death that shouldn't be allowed to continue anywhere," Mr Bullock said.
He said a solution would be for a local vet to visit the pound once a week to inject dogs to put them down.
"It's quick and less stressful for the animals, our staff or people dealing with this disgusting gas process."
He said it was time council "tightened the leash" on irresponsible dog owners.
"We put down 25-35 dogs a month and it isn't decreasing. That tells me we still have far too many reckless dog owners breeding mongrels in our district.
"They don't care if their animal becomes a roaming, dangerous or unwanted dog and one that cannot be given a new home," Mr Bullock said.
He believed an arrangement could be reached with a local vet to euthanase the dogs by injection.
"It's been suggested to me that council could do a deal with the drug company that supplies the drug and get it at a cheaper rate," he said.
To view or sign the petition, go to: http://chn.ge/1gdNxk4