Dusty is the first dog to have tested positive for the virus outside a laboratory environment.
Testing indicated that Dusty, aged two, had been infected by the virus at some stage but had recovered.
However, he still carried antibodies and there was a risk he could pass the infection on to other animals.
In a statement released through a local MP, the family said:
"Our beloved dog Dusty was humanely euthanased by our family vet after results of his most recent blood tests confirmed that he carried Hendra virus antibodies, which meant he was able to shed the virus to other animals.Dusty was the subject of campaigns through social media to have his life spared, with some suggesting he may be of more use to scientists alive than dead.
"We are devastated by the loss of our fourth family animal to the Hendra virus and are particularly saddened to witness the effect the loss of our young dog has had on our children.
"We remain in awe of the support our family has been offered by friends and strangers nationwide. The campaigns and messages of support continue to comfort all of us."
It is understood that while testing of cats and dogs on horse properties with Hendra-infected horses had been voluntary until now, authorities were considering making it compulsory.
Fifteen horses have died from Hendra in Queensland and New South Wales in less than six weeks.
It is the worst series of cases since the virus was first identified in 1994, and scientists are working to find out why.
The virus is able to infect horses and is then able to infect humans.
Of the seven known cases of Hendra in humans, four have proved fatal. The last two victims were equine veterinarians.