The victims are Italian-born Leo Biancofiore, 61, and Donata Dimase, 58.
Mr Biancofiore’s body was found at around 6:40 pm on Wednesday and his wife was taken to the Royal Melbourne Hospital with serious but non-life-threatening injuries, police said.
The dog - believed to be an American Staffordshire Terrier - remained inside the house for hours after the incident as police fired shots to stop it attacking again and a ranger attempted to capture it.
Senior Sergeant Glenn Parker said it was not clear what caused the attack.
“The dog belongs to the occupant’s son; the dog’s been here for quite some time,” he told reporters at the scene.
“It’s an older dog, it’s quite familiar with all of the members of the family and my understanding is it’s out of character for the dog.”
Senior Sergeant Parker said Mr Biancofiore’s daughter-in-law gave consent for the dog to be put down.
The dog was reportedly taken away in an animal control van about four hours after the attack.
Neighbours described Mr Biancofiore as a “nice and lovely” man who “spoke to everyone”.
Local woman, Carmen Baldwin, was cited by the ABC as saying the news came as a shock to the neighbourhood.
“It’s very sad; he’s a lovely bloke,” she said.
“He was a beautiful man, very kind, he always waved at me, very nice bloke.”
Neighbours said Mr Biancofiore had been unwell for some time and used a wheelchair and a mobility scooter to get around.
RSPCA Victoria’s Tegan McPherson said there were several causes of dog attacks, but breed wasn’t necessarily one of them.
“Breed alone is not a reliable predictor of aggressive behaviour,” she told ABC Radio.
“From what we’re hearing, this dog didn’t necessarily have a background of aggressive behaviour until this incident.
“Without understanding the context of this particular incident, it could be fear related, it could be something else going on in the environment or it could be related to pathological changes in the dog’s brain or related to pain.”