Scott Morrison has taken on the health portfolio while Greg Hunt is off sick with a bacterial infection.
Mr Hunt spent the night under observation in hospital as doctors pumped him with fluid and antibiotics.
He is expected to make a full recovery.
"Overnight, testing confirmed Minister Hunt's diagnosis to be cellulitis, a bacterial infection in his leg," his office said in a statement.
"The minister is improving and will be discharged in the coming days and expects to be back at work next week."
Australian Medical Association president Omar Khorshid dismissed any link to the vaccine.
"We can say fairly definitively that cellulitis, which is a skin infection of in this case his leg, is not going to have anything to do with the vaccination in the arm a couple of days earlier," Dr Khorshid told Sky News.
He said being vaccinated was the best possible health decision Australians could make during the pandemic.
The prime minister is confident the minister will be back in time for parliament next week.
"He will be fine by next week, he will be back up on his feet," Mr Morrison told reporters on Wednesday.
"Minister Hunt and I have worked hand-in-glove over this last year in particular on this matter and until he returns I will be personally addressing the ministerial responsibilities of health and aged care, together with minister (Richard) Colbeck."
Trade Minister Dan Tehan has spoken to his cabinet colleague and is confident Mr Hunt will bounce back.
"He was the normal vibrant, bubbly Greg and he's as keen to get back to work as he possibly can be," Mr Tehan told the ABC.
He was not as definitive when asked whether Mr Hunt would make it back for parliament.
"I'm not quite sure, it will obviously depend on the speed of his recovery, he's having a series of antibiotics and my hope is he'll be back with us next week," Mr Tehan said.
"But in the meantime, the prime minister has stepped up to the mantle and has taken on the health portfolio, so I think we can all agree that we're in very safe hands while Greg is recuperating."
There are now three federal cabinet ministers on sick leave.
Attorney-General Christian Porter and Defence Minister Linda Reynolds have both taken indefinite leave after being caught up in separate rape allegations.
Victorian Premier Daniel Andrews was also admitted to hospital on Tuesday after suffering broken ribs and damaged vertebrae in a fall.
Mr Hunt was joined by former prime minister Julia Gillard and Health secretary Brendan Murphy in Melbourne on Sunday, when the three became some of the first Australians to receive the AstraZeneca jab.
At the time, Mr Hunt said vaccinations safeguarded the community from more than just coronavirus.
"Whether it's smallpox, measles, mumps, rubella, whooping cough, influenza, and now COVID-19, vaccinations can save lives and protect lives," he told reporters.
Mr Hunt, who has served as health minister since 2017, has been overseeing Australia's medical response to the coronavirus pandemic.
Australian Associated Press