Sarah Darnley, who was 45, was among four passengers who died when the helicopter crashed around two miles west of Sumburgh Airport.
The other three victims named were Duncan Munroe (46) from Bishop Auckland, Gary McCrossan (59) from Inverness and George Allison (57) from Winchester.
Sarah Darnley is the first woman to be killed in a UK offshore incident.
Police have reported that three bodies were recovered from the sea and work is under way to recover the body of the fourth victim. The 14 survivors from the crash were taken to Gilbert Bain Hospital in Lerwick with five having later been discharged, the remaining nine have been kept in hospital for observation or for the effects of exposure.
Assistant Chief Constable of Police Scotland, Malcolm Graham said: “Our sympathies are very much with the families of those affected at this difficult time. All families have been informed and specially trained family liaison officers are currently providing support.”
Lifeboat crews and rescue helicopters from Shetland and RAF Lossiemouth attended the accident, which happened at around 6.30pm on Friday evening.
Earlier an RNLI spokesman said: “Obviously this is the news that everyone, including our lifeboat volunteers, dreaded. Our thoughts and prayers are with the families and loved ones of those concerned.
“We can also confirm that one of our lifeboats has been involved in reclaiming wreckage from the scene as part of the operation.”
The helicopter was operated by CHC who were flying for the Total oil company. It lost contact with air traffic control while approaching the airport, which is on the south tip of Shetland.
Investigators are already on the scene of the accident, which is the fifth involving Super Puma helicopters over the North Sea in the last four years.