In a statement late on Tuesday, Mr Morrison said he was experiencing flu-like symptoms and would be recovering over the next week while working from home in Sydney.
"I had tested myself daily since Sunday, including (Tuesday) morning, with all tests returning a negative result," he said.
"I took a further test (on Tuesday) evening after developing a fever. The test was inconclusive so I took a PCR test which returned a positive result."
Ben Morton, the minister assisting the prime minister and cabinet, also announced on Wednesday in a statement he tested positive to COVID-19.
"I am not experiencing any significant symptoms, only a mild headache," he said.
"I will continue to fulfil both my ministerial and electorate responsibilities while isolating in Canberra, before returning to Western Australia."
In a video posted to Facebook on Wednesday morning, Mr Morrison thanked people for their support.
Opposition Leader Anthony Albanese said he had messaged the prime minister since the positive COVID result.
"All Australians wish him a speedy recovery," he told reporters in Brisbane.
"On behalf of the Australian Labor Party, we certainly wish the prime minister to be back on deck and fully fit and healthy as soon as possible."
Defence Minister Peter Dutton, who was alongside the prime minister during a lengthy press conference in Canberra on Tuesday, has tested negative.
Mr Morrison plans to virtually chair upcoming meetings of the cabinet's national security and expenditure review committees using virtual technology.
He remains focussed on the floods crises in Queensland and NSW and the Russian invasion of Ukraine.
"(The prime minister's) determined to work from home, and he's got a secure video network set up there," Mr Dutton told Sky News.
"He's in good spirits."
Queensland Premier Annastacia Palaszczuk, who was with Mr Morrison in Brisbane earlier this week, said she hoped he recovered soon.
"It just goes to show that COVID is still out there," she told ABC television.
"Thankfully, when I met with him it was very short and we both had our masks on. In our job, we meet lots of people all the time."
Nationals senator Matt Canavan said COVID-19 had taught almost everyone how to do their job virtually.
"There is no perfect time for these things to happen, but we've all got used to over the past couple of years doing things via Zoom and other technologies," he said.
"I'm sure the PM will do everything he can remotely to coordinate, help and assist."
Mr Morrison cancelled plans to join NSW Premier Dominic Perrottet on Tuesday night to watch the Opera House being lit up in the colours of the Ukrainian flag.
He had met Nauru President Lionel Aingimea in Sydney earlier on Tuesday.
Jenny Morrison and their two daughters have tested negative and, as close contacts, will isolate for seven days.
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