Albanese was evacuated from his home at 18:00 local time (07:00 GMT) and taken to another location for several hours, police said.
One of the emails, which was written in Chinese, said large quantities of nitro-glycerine explosives have been placed around the Lodge, the name of the prime minister’s official residence in Canberra.
“If you insist on proceeding with the performance, then the Prime Minister’s Lodge will be blown into ruins and blood will flow like a river.”
The threat was reported to police by Shen Yun on Tuesday, according to the Australian Broadcasting Corporation.
A statement from Shen Yun sent to the BBC said they would proceed with their performances in Australia and called on the Australian government to conduct a comprehensive investigation.
Asked about the email threats, a spokesperson for the Australian Federal Police declined to comment.
In a social media post on Wednesday morning, Albanese thanked police and people who had sent in “kind messages”, accompanied by a photo of his cavoodle guarding the door.
“Toto on alert but all good,” he said.
Interrupted by protesters at an event on Wednesday, Albanese also repeated previous calls to lower the temperature of political debate in the country.
“I think it’s just a reminder to take every opportunity to tell people turn the heat down, for goodness sake, we can’t take these things for granted.”