The British Airways flight was halfway between Miami and Heathrow when the message was played by accident at about 3am.
Many of those who were awake and heard the announcement began screaming. The plane was cruising at about 35,000ft at the time.
Cabin staff quickly reassured passengers that the message had been played by accident.
Last night a father from Edinburgh, Scotland, who was onboard, said: "It was about 3am. An alarm sounded and we were told we were about to land in the sea. I thought we were going to die.
“My wife was crying and passengers were screaming. Then they played an announcement telling us to just ignore the warnings."
Another passenger said: "When we landed they were handing out letters apologising, but it was the worst experience of my life. I don't think BA should get away with this."
A BA spokesman said of the scare on flight BA 0206 on Friday: "The cabin crew cancelled the announcement immediately and sought to reassure customers that the flight was operating normally. We apologise to customers for causing them undue concern."
Last year, a similar message on another BA flight from Heathrow to Hong Kong also announced that the plane was about to crash.
An automated female voice played to passengers onboard the jet said: "This is an emergency announcement.
"We may shortly need to make an emergency landing on water."
The 275 passengers flying over the North Sea prepared for the worst, until cabin crew ran down the aisles to say it was all a mistake.
Michelle Lord, 32, of Preston, Lancs, said: "People were terrified, we all thought we were going to die. They said the pilot hit the wrong button because they were so close together."
Another traveller said: "I can't think of anything worse than being told your plane's about to crash." BA apologised for the "undue distress".
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