Julie Hounsom has had to undergo reconstructive surgery on her badly mauled hand following the attack.
The 43-year-old was injured after being bitten by the animals while walking her boxer dog Bruan at 1pm on Wednesday. Police have captured two dogs believed to be responsible and they have been moved to secure accommodation.
The owner has also been traced and a report sent to the procurator fiscal, although no charges have been made.
It is believed the animals are behind other attacks in the area on dogs and their owners, including one incident where a puppy was killed.
Ms Hounsom watched in horror as the animals began savaging her two-year-old pet, repeatedly biting and clawing at Bruan, in Maybury Drive, East Craigs.
She tried desperately to scare off the bull terriers, punching and kicking them, before being forced to prise open the jaws of one animal who had sank its teeth into the terrified boxer. But the bull terriers turned on her and savaged her hand, leaving her with severe injuries before the canine attackers fled.
Her nephew, Sam O'Brien, said his aunt, who was taken to Edinburgh Royal Infirmary, had been left in shock by the incident. The 19-year-old IT worker, of Silverknowes Crescent, said: "These dogs just appeared out of nowhere and started mauling Bruan. My aunt tried to stop them and they went after her.
"Her hand is really bad. The doctors had to clean out the wound first before surgery. She's still suffering from shock."
Ms Hounsom, who lives alone in East Craigs, was too upset to talk about her ordeal.
Bruan underwent surgery yesterday after suffering from serious bite and claw wounds, but is expected to recover.
Mr O'Brien added: "The police told us that they were hunting for the dogs after being contacted by other dog owners about attacks.
"These kind of dogs go for the throat so it's fortunate that a young child or someone hasn't been killed."
A police spokesman would not confirm whether the animals were responsible for other attacks.
He said: "The dogs are currently in secure accommodation. A report will be submitted to the procurator fiscal."
Jo Wilson, spokeswoman for the Scottish SPCA, said: "Owners have a legal responsibility for their dogs in public places.
"It's a shame that Staffies were involved in this case because they have a bad reputation. They are unfairly seen as macho dogs and not socialised properly so they aren't used to being around other dogs."
"We believe in punishing the deed, not the breed. The owners have to exercise the responsibility of ownership.
"Unfortunately, these dogs can be very strong and can be trained to be very aggressive."