All 12 passengers, including members of the group, were injured but none of their lives were in danger, Uri cantonal police said.
The accident occurred between Erstfeld and Altdorf at around 11am when the bus, carrying two drivers and 11 band members, flipped on its side for reasons not immediately known.
The bus, equipped with sleeping berths, was taking members of the band from Monaco to Hengelo in the Netherlands, police said.
No other vehicle appeared to be involved in the crash.
The injured passengers were transported to the Uri cantonal hospital, as well as other medical facilities outside the canton, police said.
In addition to police, fire fighters from Altdorf, the Rega emergency rescue team and ambulances from Uri and Nidwald responded to the accident scene.
Precise information about injuries Miller, 53, suffered were not immediately available.
The Grammy Award-winning bassist is known for his work with such other jazz giants as Miles Davis and Herbie Hancock.
He recently performed in Zurich for the jazznojazz event on November 1st.
In the last entry on his website, Miller writes about taking a few days off before a scheduled performance in Monte Carlo on Saturday, from which the band members were evidently returning.