A Stagecoach bus from Merthyr Tydfil ended up in front gardens off the A470 North Road at 1340 BST on Saturday.
The casualties - a mix of those travelling on the bus, including the driver, and in the car - have been taken to hospital.
The X4 bus had 20 passengers on board. The main road north of the city centre was closed in both directions.
One person is believed to have suffered serious injuries while the others sustained minor injuries.The front of the bus was badly damaged in the crash
The single-decker X4 bus was travelling from Merthyr to Cardiff when the crash took place between Colum Road and Maindy Road.
South Wales Police say the road is closed in both directions between the Gabalfa interchange and Colum Road.
They advised motorists to stay clear of the area.
BBC Wales reporter Kate Scott-Williams, who spoke to police at the scene, said a total of nine people were believed to have been injured.
"There's only two vehicles involved, a car and a bus. Both vehicles were travelling towards the city centre," she said.
"There was a collision of some sort and they then veered off the road into a garden - they've ended up across two gardens."
The road was still closed at 1730 BST for crash investigations to continue.
A spokesman for Stagecoach said: "A car was in collision with our X4 service around 1.40pm today in North Road, Cardiff.
"The single-decker bus, which was travelling from Merthyr to Cardiff, had around 20 passengers on board."
He added: "Our first thoughts are for the welfare of those who have been injured. We will also be assisting the police in any way that we can with their investigation into the accident."
Cardiff Bus said the closure of North Road was seriously delaying all services to the north of Cardiff.
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