Greenpeace and Nature and Youth disputed the Oslo District Court's verdict last month, particularly that Norway could not be held responsible for greenhouse gas emissions from the use of its oil and gas exported abroad.
The Oslo court said a 2015 licensing round that granted offshore exploration rights to companies including Statoil, Chevron, Lukoil and ConocoPhillips was acceptable under Norway's anti-pollution laws.
"There is already enough carbon dioxide in the atmosphere to seriously damage our future," said Truls Gulowsen, head of Greenpeace Norway, in launching the appeal. He said Norway's oil and gas, when burnt abroad, was stoking global warming.
Norway's attorney general Fredrik Sejersted said his office would comment only once it had received and carefully studied the appeal.
Greenpeace says the court battle is the one of the first in the world to test whether climate change is a threat to rights enshrined in a nation's constitution.
About 100 national constitutions around the world, including Norway's, include guarantees of a safe environment.
Cathrine Hambro, a lawyer for the greens, said the 2015 Paris climate agreement among almost 200 nations added weight to the lawsuit with its goal of ending the fossil fuel era this century. The pact, however, lets each country set its own goals.
On January 4, the Oslo District court ruled against the two groups and also ordered them to pay the state's legal costs of 580,000 Norwegian crowns ($75,000).
Gulowsen said the greens' legal fees, before accounting for the state's costs, totalled 3 million crowns, with about 1.5 million raised in donations. The next round could cost the same, for a total 6 million.
Greenpeace says the money shows the case is far more than a publicity stunt.
The two groups appealed directly to Norway's Supreme Court, hoping to bypass the Appeals Court. If the Supreme Court refuses to hear the case, it would go to the Appeals Court instead.
Norway is western Europe's largest producer and exporter of oil and gas and is looking ever further north. So far in the Arctic, it produces oil at Eni's Goliat field and gas from Statoil's Snoehvit.
Conservative Prime Minister Erna Solberg has said Norway can keep pumping for decades while complying with the Paris climate agreement.
The Norwegian Oil and Gas Association maintained its view that Arctic drilling is an issue for parliament to decide, not the courts, spokesman Tommy Hansen said.
Reporting By Alister Doyle