The HMS Dragon diverted 500 miles to reach the 13 Britons and one American after the mast and rudder had broken off their racing yacht.
The 60ft (18m) Clyde Challenger yacht had been travelling to the UK from the Azores when it was damaged in stormy conditions on Thursday.
The navy's Type 45 Destroyer rescued the crew at 14:30 GMT on Saturday.
The Clyde Challenger was 610 miles south-west of Land's End when HMS Dragon arrived on Saturday.
Chemical tanker CPO Finland, aided by RAF and US Air Force planes, had tried to rescue the Clyde Challenger's crew three times, but failed due to bad weather.
Petty officer Max Grosse, of HMS Dragon, said: "Despite racing through the night, we only had three hours of daylight remaining in which to safely remove the crew.
"The prevailing weather conditions and notorious Atlantic swell made it enormously challenging though and really tested the skills of my experienced sea boat coxswains."
The yacht, which is normally berthed in the Clyde Estuary, was designed to compete in the Clipper Round The World yacht race and had been used for corporate, private and charity charters, according to its website.
Its owner, Lewis Learning Ltd, said that the yacht could not be recovered, and that the crew of the Clyde Challenger would arrive onshore on Tuesday.
It extended "huge thanks" to all those involved in "organising and executing the safe transfer of the crew".
A Royal Navy warship has rescued 14 sailors who were stranded for two days in the Atlantic Ocean.