A second woman has been found dead inside an apartment building on the same street but it is unclear whether this is connected to the incident.
Police said the man attempted to carry out further stabbings and there were “multiple crime scenes”.
Video footage shows the suspect shouting “Allahu Akbar” and “shoot me”.
He jumps on a car wielding a knife as another man confronts him with a wicker chair.
The incident unfolded at about 14:00 local time (04:00 GMT) on Tuesday near Clarence St and King St, a busy area in the city centre.
The injured woman was taken to hospital in a stable condition. Police said there was no ongoing threat.
Police said they were investigating whether the body found in the apartment was linked to the incident.
The motivation for the attack is not yet known, said Supt Gavin Wood of New South Wales Police.