In a Twitter statement, police said they "have arrested the driver of the car. Extent of injuries are not known at this stage."
"Paramedics have now transported 13 people to city hospitals. Two more people are being assessed by paramedics at the scene," Ambulance Victoria said on Twitter.
Ambulance services added that some of the injured were in "serious condition".
Victoria police urged "any witnesses to go to the Melbourne West Police Station at 313 Spencer Street, Melbourne and all vehicular and pedestrian traffic to avoid the area."
A witness told ABC Radio Melbourne: "I saw a car, an SUV coming at high speed and really just heard the collision with people with bags and what must be shopping trolleys -- and I hope not prams [baby carriage]."