Alfie, a young Jack Russell, had a large chunk of his right side torn away and was left with deep puncture wounds to his liver, chest and leg.
He was only saved after neighbours rushed him to a vet surgery, where staff carried out a three-hour operation to stabilise him.
The bulldog's owner was bitten badly on the hand as he tried to prise his dog's jaws from Alfie's side, and drove off with his pet after he had freed him.
The attack happened in front of horrified young children, and Alfie's owners say it is only luck that the youngsters were not the victims.
They now want the dog identified and to be dealt with if it is found to be dangerous.
Alfie's owner, Jessica Orchard, was in Cyprus to be a bridesmaid at her sister's wedding when the attack happened, and had left her three-year-old dog with her neighbours in Kennard Road, Kingswood.
They were taking Alfie out to use the toilet when what is thought to be a silver Volkswagen Golf drove by.
Kim Redmond, partner of Ms Orchard's father Paul, said: "The dog flew out of the window and savaged Alfie.
"His owner stopped and had to prise Alfie out of his dog's mouth and then just drove off. He just disappeared and didn't come back the next day either.
"We think from talking to neighbours that the dog is local, and that it is an American bulldog."
Alfie's side was torn open and neighbours rushed him to Fernlea vets on Blackhorse Road. Vet Ellie Leister said: "I pulled back the towel and I could see his abdomen. It was pretty horrific and he was not in a good way at all, and there was no guarantee that we could save him."
After stabilising Alfie, a surgeon carried out a three-hour operation to stop his liver bleeding, and the following day he operated to repair his wounds. Ms Leister said: "He had puncture wounds to his right as well, and Matt the surgeon said he had never seen a dog injured so badly that had lived."
Alfie is on the mend, and is able to stand and wag his tail. But his owners are worried that such an attack could happen again. Mrs Redmond said: "Our main concern is that it could have happened to children.
"It was bad enough that it was another dog, but if it was a small child it could have been a lot worse.
"We would like the owner identified and if the dog is found to be dangerous then it needs to be addressed."
Bristol City Council dog warden Mark Coombs said: "This was a very upsetting incident where a Jack Russell has been very severely injured and a young girl traumatised."
Anyone with information about the incident should call 0117 9222500, quoting 247994.
(Bristol Evening Post - September 28, 2011)
