The ship - carrying 90 Australians - was freed by a fisheries research vessel at high tide on Thursday, said the cruise ship's owner, Copenhagen-based SunStone Ships and the Arctic Command, which had been co-ordinating the operation.
It was done "based on a pull from the vessel (owned by the Greenland government) and vessel's own power".
"There have not been any injuries to anybody onboard, no pollution of the environment and no breach of the hull," it said.
The name of the Greenland ship was Tarajoq and it belongs to the Greenland Institute of Natural Resources, a government agency.
The ship's owner added that "the vessel and its passengers will now be positioned to a port where the vessel's bottom damages can be assessed, and the passengers will be taken to a port from which they can be flown back home".
The cruise ship ran aground above the Arctic Circle on Monday in Alpefjord, which is in the Northeast Greenland National Park, the world's northernmost national park.
The park is almost the size of France and Spain combined, and approximately 80 per cent is permanently covered by an ice sheet.
Alpefjord sits about 240km from the closest settlement, Ittoqqortoormiit, which itself is almost 1.4km from the country's capital, Nuuk.
The Bahamas-flagged vessel has passengers from Australia, New Zealand, South Korea, the United Kingdom and the United States.
It has an inverted bow, shaped like the one on a submarine, 77 cabins, 151 passenger beds and 99 beds for crew, and several restaurants.
It comes as the ship's operator said on Thursday at least three passengers onboard have contracted COVID-19.
"These passengers are currently in isolation," the Australia-based Aurora Expeditions said in a statement.
"They are looked after by our onboard doctor, medical team and crew, and they are doing well."
The others on the MV Ocean Explorer were "safe and healthy", it said.
The Sydney Morning Herald quoted a retiree from Australia, Steven Fraser, who is on the ship, saying "everyone's in good spirits".
"It's a little bit frustrating, but we are in a beautiful part of the world," he told the newspaper, adding he had come down with COVID-19 on the ship.
The cruise ship had made three failed attempts to float free on its own during high tide before it was pulled free on Thursday.
Denmark's Danish Maritime Authority has asked police in Greenland to investigate why the ship ran aground and whether any laws had been violated, a police statement said, adding that no one has been charged or arrested.
An officer had been on board the ship to carry out "initial investigative steps, which, among other things, involve questioning the crew and other relevant persons on board", it said.
The ship's owner said in a separate statement the cruise liner began its latest trip on September 2 in Kirkenes, in Arctic Norway, and was due to return to Bergen, Norway, on September 22.
