Five of the prisoners were shot dead, while the sixth was burned to death, said Alfredo Munoz, the prison's director in the city of Guayaquil, who was stood down by the government after the incident on Thursday.
Fourteen other inmates were injured, the La Hura newspaper reported.
The violence broke out when the murder of a detainee by another inmate "provoked a brawl", according to Munoz.
Five of the victims have been identified, while the sixth remains unknown because his corpse was "incinerated", he said.
Four suspects were also identified and a firearm and other weapons were seized, a security official said.
Family members of the prisoners protested outside the prison to demand information about their relatives, according to images broadcast by local media.
The incident comes a month after the government declared a crisis in the country's penitentiaries.
Three inmates had escaped from the same prison on Monday, while two inmates were killed last week.
According to official figures, there are currently about 38,500 people in Ecuador's prisons which are only built to hold 28,000.
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