Witnesses said people were thrown through the air after being hit by the vehicle, which did not appear to be trying to stop as it "mowed everybody down".
Victoria state police said they had arrested the driver of the car after it "collided with a number of pedestrians" in downtown Melbourne at a busy intersection just after 4.30pm local time (0530 GMT).
A second man was also arrested
"We believe based on what we have seen that it is a deliberate act. The motivations are unknown," Victoria Police's Commander Russell Barrett told reporters in Melbourne.
Paramedics were "treating and transporting to hospital" 13 people, with some seriously hurt, ambulance officials said.
Sky News Australia reported that a pre-school child with a head injury was taken to hospital in a serious condition.
Citing witnesses, Sky said a white Suzuki Grand Vitara with two men inside drove into the crowd, with no signs the vehicle made an effort to slow down.
Local media showed photos of one man wearing a long-sleeve top being dragged from a car, while a bearded second man wearing a red checked shirt was seen handcuffed and sitting on the ground.
In a tweet, police appealed to members of the public to upload any images they might have of the incident to a cloud address to help assist with their investigation.
A witness, named only as Sue, told Melbourne radio station 3AW that she heard screams and saw "people flying everywhere".
"We could hear this noise, as we looked left, we saw this white car, it just mowed everybody down," she said.
"People are flying everywhere. We heard thump, thump. People are running everywhere."
Another witness, John, told ABC Radio Melbourne that he saw a "SUV coming at high speed".
"(I) really just heard the collision with people with bags and what must be shopping trolleys - and I hope not prams," he said.
"I've really never seen anything like this before and I haven't stopped shaking."
The intersection is one of Melbourne's busiest, a local shop owner told national broadcaster ABC, and is particularly crowded at this time of the year ahead of the Christmas break, with school holidays under way.
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