Police were called to Stonegate Road in the Meanwood area of the city at 9.54pm where the Renault Clio crashed.
A 12-year-old, two 15-year-olds, a 24-year-old and a 28-year-old died in the smash, West Yorkshire Police said, while two 15-year-olds are in police custody.
The families of the deceased have all been informed, investigators said.
Stonegate Road remains closed while the major collision enquiry team investigates the cause of the accident.
More to follow