CCTV captured four people cutting down the statue at 3.45am on Monday, the City of Melbourne says.
The council has provided the footage to Victoria Police and is assessing the damage to the statue to consider if it can be repaired.
Video posted to social media by an anonymous activist account appeared to show a person sawing one of the statue's ankles before it fell to the ground.
Victorian Premier Jacinta Allan said people needed to respect public property and the state government would work with the council if it chose to reinstate the statue.
"That's not a way to get your message across, there's much more effective ways to do that," she told reporters.
"That is through doing it peacefully and respectfully."
It's the third time a memorial Captain Cook has been damaged in Melbourne this year.
A statue at St Kilda's Jacka Boulevard was also sawn off at the ankles the day before Australia Day.
The words "the colony will fall" were written in red spray paint on a stone plinth underneath where the statue usually stands.
The same day, a statue of Queen Victoria at Queen Victoria Gardens near the city was also covered in red paint and graffiti.
A second monument to Captain Cook at Edinburgh Gardens in Fitzroy was  found broken and covered in graffiti over the Australia Day long weekend.
Melbourne's Port Phillip Council has vowed to repair and reinstate the monument at St Kilda while a decision has not yet been made about the future of the Edinburgh Gardens monument.
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