Matthew and Edward Boreham were picked up from their 24ft vessel about 900 miles south of the Azores and 21 miles from the spot where their distress beacon was washed overboard, emitting signals which sparked the search.
The brothers, who trained on the River Thames, told how they had been dogged by problems since they set out on the 3,000-mile transatlantic race organised by round-the-world yachtsman Sir Chay Blyth.
Matthew Boreham, 28, of Sunbury, Surrey, said they had really begun to get worried in the past few days. "I have never been so glad to see the rescue boats. We both feel lucky to be alive."
He told how they lost their power supply and then had to survive on their emergency water rations for 21 days after their own water supply became infected.
"We have tried to keep each other chirpy by listening to the World Service and telling jokes."
Despite delight at the rescue from their families, there was also disappointment that their decision to leave the boat would end their chance of race victory.
Their father, Tony Boreham, of Norwich, said: "It's a fantastic relief, but I can tell you now my sons will be totally gutted."
Edward, 31, had begun to fall ill in recent days, but was not thought to be seriously sick. The race spokeswoman, Claire Fraser, said both brothers were "fine". They will sail to Barbados in the rescue vessel, which will take two to three weeks.
- Louise JuryReuse content