The UK Maritime and Coastguard Agency said the rig came to grief on early Monday morning local time running aground at Dalmore near the village of Carloway.
The rig was being towed by Alp Marine Services' tug Alp Forward from Norway to Malta when it encountered severe weather. The same storm forced ConocoPhillips todelayed all helicopter flights to the Ekofisk oil field.
After the storm hit the tow line snapped, and the tug was unable to keep the legs of the rig from making landfall.
It is an ignominious end for the North Sea rig, which is now headed for the scrapyard, the first Norway-compliant floater to be junked.
No one was injured in the incident, and the MCA says there is little risk to people or property, although there is some fuel on board.
The tug remains on scene to visually monitor the rig and has been joined by the emergency towing vessel Herakles from Orkney, although it will not be possible to reconnect tow lines until sea states settle.
Dutch contractor SMIT Salvage has been dispatched to aid in the recovery.
"Transocean and ALP Marine have established their emergency response rooms, SMIT salvage has been mobilised to deal with the incident. The MCA's counter pollution branch and Secretary Of State's Representative for Salvage and Intervention are monitoring the situation. Local authority and Marine Scotland have been notified," the MCA said.
The Transocean Winner is capable of operating in some 460m of water.
Built in 1983 and upgraded in 2006, the rig had just completed an 11-month contract in July with Marathon Oil Corporation that paid the rig owner $498,000 a day.
The 17,000-tonne rig, famously made the landmark Avaldsnes discovery off Norway for Lundin Petroleum in 2010, later found to be part of the giant Johan Sverdrup field.
In 2011 communication between the Aldous and Avaldsnes oil discoveries in the North Sea was confirmed, at up to 1.2 billion barrels being one of the 10 largest discoveries in the North Sea.
Despite its former lucky strike, the rig was en route to be stripped down in Malta before scrapping in Turkey.
Transocean said the costs involved in its next five-yearly class renewal survey no longer made sense given the lack of demand for rigs globally.
Transocean has seven rigs in the North Sea and 29 rigs stacked and one rig idle globally.
Transocean was famously the owner of the Deepwater Horizon, which exploded in the Macondo-1 explosion in 2010 killing 11 and causing the worst offshore accidental oil spill in the history of the industry.