The four-strong gang at first tried to outrun their pursuers. However, they were forced to give themselves up after the specialist snipers on board the Lynx helicopter disabled their boat.
A team from Cumberland then boarded the vessel and arrested the crew, who offered little resistance. The operation, which was supported by the US Coastguard, took place late on Friday night.
Defence Secretary John Reid hailed the operation as a "great success" for the Royal Navy and said it had dealt a "sledgehammer" blow to the traffickers.
"Drugs are a scourge of civilised society, crippling the lives of millions around the world," he said. "Taking the fight to those who defy the international community is as vital now as it has ever been.
"For that reason, I pay tribute to the crew of HMS Cumberland, whose professionalism and commitment to the task has dealt a sledgehammer blow to the drug traffickers."
Captain Simon Ancona, HMS Cumberland’s commanding officer, said: "The ship’s prompt reactions allowed the appropriate and lawful action to take place with the minimal use of force. We worked closely with our allies as part of an international collaboration against drug smuggling."
Cumberland, based at Devonport in Plymouth, is currently on a four-month deployment in the Caribbean to provide disaster relief and conduct anti-drugs smuggling patrols. She is supported by the Royal Fleet Auxiliary (RFA) ship Wave Knight.
In 2003, she and Wave Knight were involved in another anti-drugs operation when they seized 3.6 tonnes of cocaine in the mid-Atlantic.
