There was also one death - a man in his 70s - taking the state toll to 811 and the national figure to 899.
It's the sixth straight day of cases in the double figures as Melbourne residents anxiously await confirmation that some restrictions will soon be eased.
Melbourne's 14-day rolling new case average has risen back to 10, from 9.9, and the city's mystery cases for September 27 to October 10 are also up, by two to 13.
The regional figures remain steady at 0.4 and none, respectively.
Of the 12 new cases, seven are connected to the Box Hill Hospital outbreak, including four from the same household.
Two are related to the Chadstone outbreak and two are related to seperate aged care facilities.
One is a Geelong case that is being reviewed because it might be a residual infection.
Premier Daniel Andrews has conceded Victoria's roadmap out of lockdown is likely to be redrawn and restrictions may be eased with daily COVID-19 figures in the teens.
On Tuesday, Victorian Chief Health Officer Brett Sutton defended the state remaining in tighter restrictions than the rest of the country.
"If we had started with a few ... cases that were not really complex households, that were not really challenging circumstances, we might have been able to approach this very differently," he said.
"What we had was 20,000 cases over several weeks that have come down to a low level, but the remaining chains of transmission in Victoria are some of the trickiest, I imagine, in the world."
On Tuesday, the state government announced a $250 million package to recruit 4100 tutors.
They will work in schools next year to help more than 200,000 students who have fallen behind in their learning because of the state's coronavirus lockdowns.
"This is the single biggest boost to individual learning in our state's history," said Education Minister James Merlino.
Retired teachers and casual relief teachers will be called back for the tutoring roles, Mr Merlino said.
He added data showed one in five children had fallen behind during online learning this year.
Meanwhile, the premier is expected to face a no-confidence motion as state parliament resumes on Tuesday.
Victoria's most senior bureaucrat Chris Eccles quit on Monday after phone records revealed he spoke to the head of police as the state's hotel quarantine programme was hastily set up on March 27.
Mr Eccles is the second key public official to quit amid the inquiry into the botched programme, with former health minister Jenny Mikakos resigning last month.
The flaws in the hotel quarantine system are believed to be responsible for the state's second wave of COVID-19
Opposition leader Michael O'Brien has called on Labor MPs to back the motion and cross the floor, saying it is "quite clear" that privately, many of them are critical of the government.
But Greens MPs said they would not support the motion, labelling it a stunt.
The opposition only has one chance to move a motion of no confidence in the premier each parliamentary term and will have to wait until after the 2022 state election if it fails, assuming Labor is still in power.
