Two people have died and 11 are missing and presumed dead.
Survivors told Mexican media that their home-made raft sank as it hit the reefs surrounding Isla Mujeres on Friday morning.
The refugees said they headed for Mexico in the hope of joining relatives in the United States.
The accident happened some 150 metres (140 yards) off Isla Mujeres, in an area known for its strong currents, the Mexican Navy said.
Ten Cuban refugees managed to swim to safety.
But they said the others were taken away by the currents or fatally injured as they were pushed towards the rocks by the big waves.
“One of them head a head injury and died on the beach. The other one, a woman, was taken to hospital but also died of her injuries,” said a Navy official, Ismael Gonzalez Gil.
There were no children among the 23 Cuban refugees.
The survivors said group’s intention was to reach the Mexican coast, which lies some 220 km (140 miles) west of Cuba, and eventually cross the border into the United States.
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