The crash - which took place at around 8pm on Thursday (3pm local time) - happened in a south-eastern city of Guantánamo, around 1,000 miles away from the capital Havana. The region is mountainous with winding roads.
Officials were working to identify the dead.
Meanwhile, five of the injured were said to be in critical condition in hospital.
CNN's Patrick Oppmann reported from Havana and Hira Humayun from Atlanta.
Citizens from the UK, Canada, France, Mexico and the Netherlands were reportedly on board.
Witnesses told state-run Cuban News Agency that the driver had tried to overtake another vehicle but lost control on the wet road, resulting in the bus flipping over.More news: First photo of Jayme Closs released since she was found
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The dead ranged in age from 32 to 67, according to a tweet by Radio Guantánamo, which identified the injured US citizen as Allison Alida Kemmer.
A Frenchman, a German woman and two Argentine women were among the dead, along with three Cubans, Venceremos newspaper said.
The Cubans were two men, aged 32 and 47, and a 34-year-old woman.
As well as the Britons, passengers' nationalities are listed as Dutch, French, Mexican and Canadian.
The incident was the fourth major bus accident in a month in Cuba, which has narrow, poorly lit, pothole-littered roads.
Last year, there were 750 deaths and 7,999 injuries from 11,187 accidents on the country's roads.