New Zealand-born Cooper moved to Australia as an early teen and has earned 71 test caps with the Wallabies but his citizenship applications have been repeatedly declined.
His starring role in the Wallabies' 28-26 win over world champions South Africa on Sunday triggered calls on social media for the government to give the 33-year-old citizenship.
And on Tuesday, Australia's immigration minister Alex Hawke said authorities had altered eligibility rules which would ease restrictions on overseas-based applicants.
Cooper thanked local lawmaker Kristina Keneally, who was vocal in her support for his citizenship bid and pressured Hawke to resolve the issue.
"The Australian public put on a lot of pressure and also the media put a lot of pressure on the government to take a look at not only my case but a lot of other people who sort of fall in the same position," Cooper said.
Cooper had been playing rugby in Japan before recently being recalled to the Wallabies squad. He played his first test in four years on Sunday, booting the match-winning penalty in a perfect night off the tee.
Cooper's most recent application was rejected because he provided no evidence that he had "engaged in activities of benefit to Australia," the Home Affairs department said.
Immigration minister Hawke said current criteria was blocking the path to citizenship for worthy candidates.
"The unique work and travel demands on some of our most highly distinguished prospective Australians should not preclude them from making the cut," he said.
Cooper had responded to his citizenship rejections with humour, posting his most recent rejection letter on social media in July.
However, he admitted he was concerned at one stage about his ability to return home to Australia as the country closed its borders amid the Covid-19 pandemic.
"It hasn't truly affected me in any way apart from when I was obviously playing overseas in Japan," he said.
"I (had) the situation around the world with Covid and people not being able to come and go as they please from their home country. And this is a place where my family and I've lived for over 20 years.
"That was quite a daunting feeling."
Local media said Cooper could be granted citizenship as early as this week, meaning he could sing the national anthem as an Australian on Saturday before taking on the Springboks at Lang Park in Brisbane.
Cooper said he probably would not feel any different.
"I've said all along, there's more to being Australia than just a piece of paper."