US Coast Guard petty officer Jennifer Robertson said: “The search has resumed.”
Dan Carpenter, son-in-law of Steve Warren, one of the missing men, said: “We are holding out hope. We are aware that it is still a long shot but while there is some hope, we are concentrating on that.”
Prime Minister David Cameron said: “My thanks to the US Coastguard, which has resumed its search for our missing yachtsmen.”
The crew of the 40ft (12m) Cheeki Rafiki ran into difficulties about 620 miles (998km) east of Cape Cod in Massachusetts last Thursday while returning to the UK from a regatta in Antigua. Contact with the yacht was lost in the early hours of Friday when they diverted to the Azores.
The coastguard, Canadian aircraft and three merchant vessels searched for them throughout Friday and Saturday but called off efforts on Sunday at 5am local time amid treacherous weather.
Relatives of the four men - experienced captain Andrew Bridge, 22, from Farnham, Surrey and crew members James Male, 23, of Southampton, Steve Warren, 52, from Bridgwater, and Paul Goslin, 56, of West Camel - have been pleading with the US Coastguard to resume the search and remain convinced that their loved ones are alive.
Andrew Bridge’s grandmother Valerie said: “We are delighted. It is at least something and that is all we were asking for, all we wanted was another search.
“It might not come to anything but people want them to do it and they are trying. It seemed too quick, just two days and we were saying ‘if only they could do it (search) for a bit longer’. You never know what could happen.”
The families of the sailors have travelled to London to meet foreign minister Hugh Robertson.
Leading figures from the yachting world are among more than 175,000 people who have signed an online petition supporting the campaign to resume the search.
Health Secretary Jeremy Hunt, the local MP for one of the missing men, had appealed to the US Coast Guard (USCG) not to give up and entrepreneur and adventurer Sir Richard Branson had called on vessels near the area to keep a lookout.
Dame Ellen MacArthur, who twice broke the world record for fastest solo circumnavigation of the globe, said there was “every chance” that the sailors were still alive.
She said: “It seems that there is an element of everyone working in the dark here as the incident happened so far from land and there is no longer contact with the crew. However there is every chance that the sailors could be alive either inside the hull of the vessel, or in the life-raft which is designed to keep people alive at sea.
“There are examples of both types of survival, and in both cases for extended periods of time.”
Mike Golding, one of the few yachtsmen to have sailed around the world non-stop in both directions, had argued for a further search, especially to locate the hull and establish whether a life raft had been released.
He said: “It’s hard to imagine four men, well-equipped, even with a boat in trouble actually disappearing like that.
“The boat had all the right safety equipment, people are just saying the search period was very short.”
Some 4,000 square miles (10,360 sq km) were scanned for the vessel’s two personal location GPS beacons until no more transmissions were received from the small devices, which have a short battery life.
On Saturday, a cargo vessel which was helping with the search spotted and photographed an overturned hull which matched the description of the Cheeki Rafiki but reported no signs of people on board or a life raft.
Mr Golding said: “From the images, the yacht has lost the keel, initially they were sinking, taking on water. One imagines they put out the Mayday, prepared themselves for sinking, then the keel fell off, maybe the boat rolled over fast and the question is what happened at that point? Were they able to launch the life raft at that point?”
He added that efforts should be made to locate the hull, which he believed could still be afloat, having lost its heavy keel, to establish whether the life raft was still in place and if the sailors were gathered in an air pocket underneath.