"Last night, the police decided the time was right, and we arrested the two activists," Greenland deputy police chief Morten Nielsen told AFP.
"They are being transported to (Greenland capital) Nuuk, and it will be determined whether they they will go before a judge," he added.
Greenpeace also said the two 25-year-old activists, Luke Jones of Britain and Hannah Mchardy of the United States, had been arrested after hanging in a "survival pod" beneath the 53,00-tonne "Leiv Eiriksson" platform for four days.
"Our climbers are in jail now, but this won't stop us opposing the madness of drilling for oil that we can't afford to burn and in a region where a spill would be almost impossible to clean up," Greenpeace international oil campaigner Ben Ayliffe said in a statement.
The two activists had arrived by rubber dinghy early Sunday at the 53,000-tonne platform, which is due to begin drilling for oil 180 kilometres (110 miles) off Greenland for Scottish company Cairn Energy.
By hanging their survival pod, suspended from the underside of the rig, "25 metres (82 feet) over the freezing Atlantic Ocean," near the so-called "moon pool where the drill-bit would normally be sunk, the activists prevented the rig from drilling," Greenpeace said.
Then, "just before midnight last night, local time (0000 GMT Thursday) ... a climb team operating from the rig broke into the pod," the organisation said, adding "Danish navy inflatable speedboats were positioned below the climbers."
Deputy police chief Nielsen said the operation was run by local police in Greenland, which is a semi-autonomous Danish territory, but that the Danish navy had been on sight in case rescue efforts were needed.
"When you're suspended that high above the freezing water it's good to have some help in case," he said.
Cairn is meanwhile seeking an injunction against Greenpeace and want the group to pay fines of two millione euros (2.9 million dollars) for each day it disrupted drilling, according to a separate Greenpeace statement.
"Cairn is trying to use a legal hammer to shut down our campaign to kick the oil companies out of the Arctic, but we'll challenge Cairn and its lawyers every step of way," Ayliff said.
The company did not immediately respond when asked to confirm the legal action, but previously rejected in an email to AFP suggestions it had pressured Greenland authorities to put an end to the Greenpeace action, although it said it was not sad it was over.
"Cairn respects the rights of individuals and organisations to express their views in a safe and peaceful manner but would be concerned with anything that presents a safety risk for those involved and the operations," the company said in a statement.
The two activists could meanwhile face charges under Greenland's immigration laws, Nielsen said, adding it was unclear how quickly they would be presented before a judge.
"Under Greenland immigration laws, we have a right to detain them for 72 hours before they are presented in court," he explained.
Ayliff, who was on the Greenpeace shop Esperanza near the rig and had seen the operation, insisted that despite the arrests of Jones and Mchardy, "this isn't over."
"We must keep on pushing till the oil companies get out of the Arctic," he said in the statement.
"We stopped this rig from drilling for four days, which was four days in which a Deepwater Horizon-style blow-out couldn't happen in this beautiful and fragile environment," Ayliffe said.
Cairn meanwhile said insisted in its statement that wherever it is active it "seeks to operate in a safe and prudent manner."
"The Greenland Bureau of Minerals and Petroleum has established some of the most stringent operating regulations anywhere globally," it added.
Greenland is looking to oil prospecting as a way to ensure its economic independence.
The Arctic seabed is thought to hold about 90 billion barrels of oil and 30 percent of the world's undiscovered gas resources, according to the US Geological Survey.
