A man was arrested and "the situation is now contained", police said.
The incident was not believed to be terror-related, said Victoria Police Acting Commissioner Stuart Bateson.
The incident happened at the Bourke St Mall, a busy shopping area, just before 14:00 (03:00 GMT).
Mr Bateson told reporters that police believe the driver "deliberately drove into the crowd".
Footage broadcast on Australian media showed a maroon car driving erratically in circles in front of nearby Flinders Street railway station.
Former Victoria Police chief Christine Nixon said she saw an overturned pram at the scene. She told the ABC a woman approached her and asked where the child was.
"I don't know," Ms Nixon said. "The child is not in the pram."
Another witness told the BBC she saw pedestrians try to get out the way as the car mounted a footpath.
"It hit a few people there outside the offices and convenience store and sent them flying into the air, then kept driving," the witness said.
"It hit some more further up the path. I didn't know what was happening at first. People were screaming and there was a lot of sound and dust - I thought a building was coming down."
Melbourne's Royal Children's Hospital confirmed it was treating 10 patients from the incident. St Vincent's Hospital said it was treating up to six patients.
The Australian Open tennis event, being played about 2km (1.2 miles) away, was unaffected, police said.
Police requested that people avoid the city.