The car left City Rd about 9.15am, went down a steep bank into the Arthur St playground, travelled over a children's slide under construction and wove between trees, before coming to rest at the base of the memorial marking the site of Dunedin's first cemetery, near the corner of Arthur St and City Rd.
Acting Senior Sergeant Simon Mapp, of Dunedin, said the girl was not injured but was treated for shock at Dunedin Hospital afterwards.
Police were still making inquiries as to why the vehicle left the road.
It was not clear if a sleet shower just before the crash was a contributing factor, he said.
Dunedin City Council sports field and facilities officer Harold Driver said a cast iron fence around the memorial was damaged in the crash.
Council staff had taken the section of fencing to a specialist for a quote on re-casting, he said.
The memorial was not damaged.
