The driver of the bus and two passengers were injured in the smash - which closed the outbound carriageway of the Sheffield Parkway for half an hour just before 9pm.
South Yorkshire Police were alerted to the collision at the junction of Derek Dooley Way and the Park Square roundabout near to Ponds Forge.
Dashcam footage provided by a passing motorist in the aftermath shows the police vehicles at the scene of the smash, which left two passengers from the single decker bus and the bus driver with minor injuries.
Police said they were called at 20.17pm to be told of a two vehicle smash which had left the mini bus on its side.
A force spokesman said: "There were no serious injuries, just walking wounded. The driver of the single decker bus and two passengers were checked out at the scene.
"The Parkway was closed for around half an hour while the minibus was righted by the fire service.
"The road re-opened at 9.24pm."
Anyone with information is asked to call police on 101.