The Mexican government had requested their removal, WSVN-TV said.
The Cuban government said in an official statement the operation was carried out "without the least incident," but it was not disclosed where the men were taken.
Cuban officials said the group smashed their way into the embassy compound in a stolen bus Wednesday night and rushed onto the embassy roof. They threatened to commit suicide if they were approached, but there were no reports Friday of any suicides or attempts.
The government said the invaders were not politically motivated but instead were common criminals.
"None of them is really motivated by ideas or objectives of a political character," a statement said.
Gloria Abella, a spokesman for Mexico's Foreign Ministry, told the British Broadcasting Corp. the men were not political dissidents.
"There has been no request for political asylum, this is a different kind of situation," she said. "These are young people facing a difficult economic situation, like many in Latin America."
The Cuban government accused the U.S. government's Radio Marti of provoking the incident by repeatedly broadcasting news that Mexico was inviting Cubans into the embassy.
Officials in Havana said it was broadcast eight times and called it an invitation to occupy the embassy.
Foreign Minister Jorge Castaneda told Cuban-Americans at a reception in Miami Tuesday: "The doors will be open for any Cuban or Latin American who is interested in visiting Mexico or having contact with the embassy."
The Mexican consul in Miami, Manuel Rodriguez, on Thursday said the remark was not intended to provoke a takeover and was taken out of context.
Elsewhere in Miami, the Cuban American National Foundation accused the Fidel Castro regime of spreading the rumor to create an embarrassing incident.
It said Castro was retaliating for Mexican President Vicente Fox's meetings with dissidents during a visit to Havana Feb. 4.
Calls went out on Cuban-American talk radio in Miami Friday for a boycott of Mexico in retaliation for the eviction of the 21 men.
Shortly after the incident began Wednesday night, Castro spent 20 minutes at the compound, while supporters chanted "Fidel, Fidel."
At the same time the people on the roof chanted, "Down with Fidel."
A similar incident at the Peruvian Embassy in 1980 grew rapidly and eventually resulted in the Cuban boatlift which brought 125,000 Cubans to South Florida. This time the area was quickly sealed off and no one was allowed near the Mexican compound.
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