The accident occurred at 3pm in the afternoon and patients were through the hospital doors within two hours, transported by ambulances and civilian vehicles.
Eighteen Cubans and 22 tourists were on board the coach when it crashed between the eastern cities of Baracoa and Guantánamo.
According to official statistics, there were 11,187 road accidents in 2017 that left 750 people dead and 7,999 injured from a population of 11.1 million.
But witnesses told the station that the crash happened after the driver attempted to overtake a auto before changing his mind when he saw an approaching lorry.
Radio Guantánamo said it had spoken to the driver of the bus who said he lost control of the bus in wet conditions on a bendy section of the road between Baracoa and Guantánamo. The three dead victims identified so far are Cubans.
Viazul is run by the military's tourism wing and is one of the preferred ways for visitors to travel the island, the BBC's Will Grant in Havana reports.
This is the fourth major bus accident in Cuba in a month and the nation's roads are known to be lacking in light, narrow and often poorly maintained.
Previous year there were 11,187 accidents, resulting in 750 deaths and 7,999 injuries.