Officers were seen surrounding the vehicle before the man, said to be in his late 20s, was detained, shortly after 7:30am London time.
London Ambulance Service says it took two patients to hospital with non life-threatening injuries after cyclists and pedestrians were hit.
No one else was in the car and no weapons have been found so far.
Scotland Yard says at this stage, it is treating this as a terrorist incident and the MET’s Counter Terrorism Command is now leading the investigation.
The man is in custody at a South London Police Station and the vehicle is being searched.
The government’s Cobra Emergency Committee is meeting later.
A number of eyewitnesses have said the silver Ford Fiesta, which was travelling westbound, appeared to deliberately hit members of the public as it swerved into the opposite lane.
Footage released by the BBC shows the moment when the vehicle drives over a pedestrian crossing before crashing into the security barriers.
Parliament is currently not sitting.