Lured by the promsie of huge reserves of oil in their territory, 76 per cent of voters backed the plan in a referendum on Tuesday night.
As the result became clear, Hans Enoksen, Greenland's premier, punched the air and declared: "The tears are running down my cheeks. We have said 'yes' to the right of self-determination, and with this we have accepted a great responsibility."
The revised constitutional settlement sets new rules on splitting future oil revenues with Denmark, which heavily subsidises the semi-autonomous Arctic island.
It has had home rule since 1979, but under the new law it will control its mineral and oil resources and eventually take over 32 additional fields of responsibility from Denmark, including justice and legal affairs, as it becomes economically viable to do so.
Greenlandic will become the official language, but it will still be part of the Kingdom of Denmark. Full independence will require another referendum, which is unlikely to be held until Greenland is ready to live without Denmark's financial aid.
Denmark has insisted Greenlanders alone must decide when to cut the final ties between the two countries after nearly 300 years of Danish rule.
Anders Fogh Rasmussen, the Danish prime minister, said: "The Greenland people have yesterday given a clear 'yes' to self-determination within the (Danish] realm."
The plan is expected to be approved by the Danish and Greenlandic parliaments and go into effect on 21 June, the island's national day.
Greenlanders, who had queued to vote despite heavy snow and temperatures of minus 5C, celebrated in the streets of the capital, Nuuk, even before polling closed. There were some 'No' posters to be seen, but these were hugely outnumbered by those urging a 'Yes' vote.
Greenland is the world's biggest island that is not a continent, and is 80 per cent covered in ice.
It has a mostly Inuit population of 57,000, many of whom live in small, isolated villages. There are no roads between towns, and travel is possible only by boats or planes, weather permitting.
The United States Air Force operates a ballistic missile early warning site at Thule air base, some 700 miles north of the Arctic Circle.
Greenland receives yearly subsidies of 365 million from Copenhagen, about 30 per cent of its gross domestic product.
Shrimp and halibut fishing and tourism form the backbone of the economy, but the island is rich in minerals and its waters may hold vast hydrocarbon reserves, although full-scale production is decades away.
Climate change has caused Greenland's ice sheet to melt increasingly in recent years, making mineral exploration and oil drilling more feasible.
Drilling for oil and gas in the deep ocean off Greenland's west coast resumed in 2001, three decades after a previous effort failed to find petroleum. Exploration so far has been unsuccessful, but other countries in the northern region are staking their claims to natural resources exposed by the melting of the Arctic ice cap.
Denmark has agreed that Greenland would get the first 8.5 million of annual oil revenue, with the rest shared 50-50 with Denmark.
GDP per capita in Greenland is about two-thirds of the Danish average, but the suicide rate is more than seven times higher.
Greenland became a Danish colony in 1775 and remained so until 1953, when Denmark revised its constitution and made the island a province. Under the 1979 Home Rule Act, Greenland got its own parliament and government, as well as self-determination in health care, schools and social services.
Foreign and military affairs are controlled by Copenhagen, and Denmark's Queen Margrethe is the head of state.
All of Greenland's main political parties supported greater autonomy, except the small opposition Democrats, who questioned whether the island could afford to take over the more than 30 new areas of responsibility outlined in the referendum.
Opponents have also raised concerns about Greenland's social problems, such as widespread alcohol abuse and a high suicide rate among teenagers.
Greenlanders older than 15 drink an average of 11.6 litres of pure alcohol per year, the highest in the Nordics, according to official health statistics.
But the prime minister is confident. "It is time that we as a people encouraged, supported and strengthened each other to continue," Mr Enoksen said.
The percentage of Greenland's 844,000sq miles covered by an ice cap
The thickness of the ice in feet
The population of Greenland, of whom 47,000 live in its 18 towns
When the indigenous Inuit are believed to have come from Siberia
Almost a third of the population live in Nuuk, the capital
The population of Sisimiut, on the Arctic Circle, the second-largest town
The annual subsidy the Danish parliament gives Greenland
The unemployment rate. It varies depending on the seasons
When the Home Rule agreement gave Greenland semi-autonomous status within Denmark and established its 31-seat parliament