The rescue centre's search and rescue officer Geoff Lunt said the family were very lucky a yacht happened to be anchored nearby, as Beveridge Reef was 400km southeast of Niue and there were no other vessels anywhere in the vicinity.
"The skipper of a New Zealand-registered 18-metre steel ketch answered the emergency radio call from our Maritime Operations Centre within seven minutes. It was very fortunate they were anchored in the lagoon and listening to the distress channel at 2.30am," he said.
By the time the rescuers reached the catamaran it was taking on water and well aground on the reef. It was now likely to break up.
The British family had to leave with only the clothes they were wearing and their passports, he said.
They were now comfortable onboard the ketch, within the safety of the lagoon before sailing to Niue in the next few days.
"We would like to thank the crew of the ketch and all the rescue services involved in tracking vessel locations and successfully rescuing this family," said Luntt.