Mr Joyce said Depp had until tomorrow to put the pooches on a plane out of Australia. "Or we're going to have to euthanise them."
The minister added: "It's time that Pistol and Boo buggered off back to the United States."
Depp is thought to have brought the terriers into Australia when he and wife Amber Heard touched down in Queensland last month.
The latest Pirates blockbuster, Dead Men Tell No Tales, is being shot on the Gold Coast with the help of a A$20 million ($21.5 million) tax incentive from the Government.
Authorities started investigating after the dogs were "spotted" in a handbag being taken into Happy Dogz grooming salon last weekend.
A Department of Agriculture biosecurity officer has since visited the luxury house where Depp is staying. He is understood to have delivered a three-day ultimatum, but did not seize Pistol or Boo.
Yesterday, as media camped outside hoping for a glimpse of Depp and the dogs, fans began a campaign to save his pets. By late afternoon, more than 3000 people had signed a change.org petition.
"I just love dogs, and I saw the news this morning and thought it was crazy," Sydney woman Namita Sopal, who started the petition, told the ABC. "I understand that Johnny can't just bring the dogs into the country, but the dogs shouldn't be punished."
The 51-year-old actor could have brought Pistol and Boo in legally if he had followed a process that includes health checks and vaccinations for diseases such as rabies. Owners require a valid import permit and pets are put in quarantine for at least 10 days.
Mr Joyce said Australians had peace of mind because the authorities enforced strict regulations. He has launched an investigation to find out how Depp's dogs made it past customs officials.