The 19-year-old driver - of Afghan descent - was arrested and is under police guard at hospital with shoulder and hand injuries.
Police said he had a history of drug use and mental illness.
A second man, 24, was also arrested.
However, the police said there was no evidence the men were linked in any way to terrorism.
A police officer sustained hand and shoulder injuries during the arrest and has been taken to hospital.
Earlier, witnesses described what they saw as the horror unfolded.
'People were flying everywhere. We heard thump, thump, thump. People are running everywhere,' a witness told 3AW.
'If members of the public have video or images which might assist police with their investigation into the Flinders Street incident they are encouraged to upload them here,' the police tweeted.
Ambulance Victoria tweeted that it had transported 13 people to city hospitals - while two more were being assessed at the scene.
‘A pre-school aged child with a head injury has been taken to RCH in a serious condition,’ the organisation said in a statement.
The incident occurred on the corner of Elizabeth Street and Flinders Street at 4.41pm on Thursday.
Victorian Premier Daniel Andrews has called it an 'act of evil and cowardice'.