Greenpeace said four campaigners climbed aboard the Stena Don, a semi-submersible rig, which has been drilling the Alpha prospect in the Sigguk block, 175 km offshore Disko Island, West Greenland.
The Greenpeace activists suspended themselves in tents from the rig and have enough provisions for days, said spokesman Ben Stewart, on Greenpeace ship the Esperanza.
Greenland police said that the activists remained suspended by their climbing gear below the rig in the late afternoon.
Like any criminal, they can be expected to arrested and prosecuted, Morten Nielsen, deputy chief of police in Greenland, told Reuters by telephone. He declined to give any details about when or how that would happen.
Cairn has said Greenland could have billions of barrels of reserves but the BP oil spill in the Gulf of Mexico has raised fears about the risks of offshore drilling.
Greenland's harsh climate and remoteness would make capping a leak and cleaning up spilt oil especially difficult.
Greenpeace's vessel Esperanza has been in the area for over a week planning activities to prevent Cairn from completing its objectives before the end of the narrow summer drilling season.
The sea between Greenland and Canada is known as iceberg alley.
Commandoes from a Danish warship which has been trailing the Esperanza failed to stop campaigners from boarding the rig, Greenpeace said.
We caught them napping, Stewart said.
Greenland deputy police chief Nielsen said that police could have stopped the activists before they reached the rig, but chose not to intercept them because of the risks involved, both to the activists and police officers, in such cold waters.
He said the police were supported by the Danish navy but that maintaining law and order in the area was a police matter.
Cairn declined comment.
The Stena Don has been drilling the Alpha-1 well since late June. It was expected to hit target depth in 55 days and Cairn investors are expecting to hear whether it has been successful in the coming weeks.
Edinburgh-based Cairn last week completed drilling of its nearby T8 prospect. Although the T8 well failed to hit oil, the company said the results proved the existence of a hydrocarbon formation.
The Stena Forth drillship, which drilled the T8 prospect, is scheduled to commence drilling another prospect soon.
Greenpeace declined to say if it might try and disrupt that vessel's operations next.
Cairn shares ended up 0.5 percent, outperforming a 0.4 percent drop in the STOXX Europe 600 Oil and Gas index.
Cairn's wells are the first to be drilled in Greenland in over a decade. Six wells were drilled in the 1990s but failed to find oil or gas in commercial quantities. However, companies hope that better technology will now allow them to discover big reservoirs.
Big oil companies including Exxon Mobil and Chevron have bought exploration licences.
(Additional reporting by John Acher in Copenhagen)