Police in the canton of Grisons said the accident on Thursday occurred near the village of Susch, and images of the scene published online showed the vehicle lying on its side next to the road, surrounded by police and emergency service personnel.
Three of those hurt were in a serious condition, seven had moderately serious injuries, and 30 were slightly injured, police said, noting there were 45 adults on the bus.
The dead have yet to be identified.
Prosecutors and police were investigating what led to the accident, though no other vehicles were believed to be involved.
Oad, a travel agency behind the trip, said in a statement that it was "deeply shocked and devastated by the tragic bus accident" and its thoughts were with those involved and their loved ones.
Swiss President Guy Parmelin, writing on X, vowed that Swiss authorities would ensure a "comprehensive and thorough investigation" of the case.
Images from the scene showed crews in fluorescent suits cutting into the overturned bus with power saws overnight, as police looked on.
"The Graubünden public prosecutor's office is clarifying together with the cantonal police the circumstances that led to the coach accident," a police statement said.
According to initial findings, the coach crashed into a guardrail and then veered off the road to the right.
It overturned and came to rest on its side. A major deployment of police, emergency services and firefighters was on site until late at night, including helicopters.
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