Greenpeace's ship Esperanza is already in the area in protest and has clashed with a Danish warship as it approached the 500 metre exclusion zone around the rig.
Green groups say a blow out like that at BP's Macondo well in the Gulf of Mexico could cause even more damage in Greenland as cold conditions would mean limited evaporation of the oil.
Greenland, which Denmark has held sovereignty over since 1721, also has limited facilities for dealing with a major spill.
Cairn said that though the gas in Baffin Bay it found was too small to be commercially exploited, it supported the view that the area could yet yield material finds.
"I am encouraged that we have early indications of a working hydrocarbon system with our first well in Greenland, confirming our belief in the exploration potential," Chief Executive Sir Bill Gammell said in a statement.
Exploration director Mike Watts added that the gas was of a type that is sometimes found in association with oil.
The well in question – T8-1 – has not yet reached its target depth and Cairn has plans for at least two more in its current drilling programme.
However the company admitted if it suffered an accident similar to the BP spill in the Gulf of Mexico in April, it would be unable to cap a blow out for months.
While analysts said the news was "exciting, Greenpeace demanded Cairn immediately halt its operations and make public their safety procedures.
"Companies like Cairn need to leave the Arctic alone and instead work quickly to develop safe and clean alternatives that will actually help us get off fossil fuels for good," said Greenpeace activist Leila Deen, who is on the Esperanza.
"Cairn might be a step closer to finding oil off Greenland, but this takes us one step back in the fight against climate change and poses a grave threat to the fragile Arctic environment." Cairn Energy has been turned from an oil minnow to a global energy powerhouse under the stewardship of Sir Bill, a former Scottish rugby international who has long been friends with Tony Blair and George W Bush.
He met the future Prime Minister at Fettes, a public school in Edinburgh where he was his classmate and debating partners.
His interest in oil was sparked during university summers spent with the Bushes in Texas When George W Bush became president, his first words to Mr Blair were: "I believe you know my old friend Bill Gammell."
Greenland is self-governing but barring a severing of ties with Denmark, a major oil find will put the Danes in the unlikely position of becoming an oil-rich nation on a par with the likes of Saudi Arabia.
"Daily Mail"