The Ocean Explorer has 206 passengers on board but ran aground on Monday in Alpefjord, a national park 870 miles (1,400km) northeast of Greenland’s capital Nuuk.
Denmark’s armed forces and the cruise ship’s operator have said there were no injuries among the passengers or crew on Thursday.
There had been three prior attempts to rescue the cruise ship which had three cases of Covid onboard.
“These passengers are currently in isolation. They are looked after by our onboard doctor, medical team and crew, and they are doing well,” the Australia-based Aurora Expeditions said in a statement addressing the Covid cases.
The others on the MV Ocean Explorer were “safe and healthy”, it added.
The vessel left the Norwegian port of Bronnoysund on September 6 and ran aground 249 miles (240km) away from the nearest settlement, Ittoqqortoormiit.
Australian newspaper 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.”
The vessel has 77 cabins, 151 passenger beds and 99 beds for crew, and several restaurants.
Ahead of the successful rescue operation, the Arctic Command said its "first priority" was to have its larger inspection vessel, Knud Rasmussen, reach the site on Friday.