The accident occurred shortly before 8am on Friday.
Of the 18 people on board, 17 were able to exit the bus on their own, with one young woman needing assistance.
The 22-year-old was helped out of the bus by paramedics and taken to Mälarsjukhuset in Eskilstuna with neck pain, police said.
"It was not as serious as we originally worried," Anette Wilhelmsson, an officer at Örebro police, told news agency TT.
Other passengers on the bus were taken to a collection point nearby and treated by paramedics. Four, including two children, were later taken to local hospital Mälarsjukhuset, with minor injuries.
Witnesses said it was very slippery at the site of the accident.
A number of school bus services were cancelled in the region and in Västmanland due to slippery conditions, according to local media.
"We can not take a chance with children in buses," Fabian von Essen, head of Axelssons Traffic, told local media.
Slippery road conditions were reported in several areas across Sweden on Friday morning and a series of other traffic accidents were reported, but there were no serious injuries.
On highway E22 near Botorpström in the North Kalmar region south of Stockholm, a garbage truck overturned and lay on the side of the road. The driver was injured and taken by ambulance to hospital.