Emergency services were called to the collision, at the junction of St George’s Road and Eastney Esplanade, just after midday.
The driver and passenger received ‘moderate injuries’, which are not thought to be life-changing.
Speaking from the scene, an official from South Central Ambulance Service, told The News: ‘There were two casualties. One was driving the vehicle the other was a pedestrian.
‘The pedestrian hasn’t been hit by the car. Both have moderate injuries but that are not life-changing.
‘They have both been taken to Queen Alexandra Hospital for treatment.’
Eye-witnesses spoke of their disbelief as to what happened.
A 76-year-old woman, who lives in Marine Court opposite the crash site, was shocked by the collision.
She told The News: ‘It’s just terrible. It looked like someone might have been pinned by the car. The fire brigade and police arrived incredibly quickly - they were here within minutes.’
She added she had safety fears about the junction between St George’s Road and Southsea Esplanade – although there is no suggestion that speed or careless driving was a factor in this accident.
‘People come round that corner very fast,’ the pensioner said. ‘I have been crossing the road with my dog before and some boy racers have come round the corner. It’s terrifying.’
Emergency care was provided to the two people by firefighters from Cosham and Southsea fire stations, who arrived at 12.12pm.
A spokesman from Hampshire and Isle of Wight Fire and Rescue said: ‘No-one was trapped. Firefighters provided immediate emergency care to one person who had got themselves out of the vehicle and one pedestrian who had fallen nearby.
‘Crews left after handing the incident over to police.’
Both lanes on Eastney Esplanade were closed by following the crash, as officers worked to clear the route.
A spokeswoman from Hampshire Constabulary said this afternoon: ‘No serious injuries have been reported at the current time.
‘We are at the scene and the road is currently closed.
‘We would urge road users to avoid the area and take alternative routes while the road is being cleared.’