In early January, the Oslo district court dismissed a first case by the Norwegian branch of Greenpeace and Natur og Ungdom (Nature and Youth) which had sued the government for granting exploration licenses in the Barents Sea in May 2016.
"It is crystal clear that the state is violating the Constitution and our right to a healthy environment by allocating new oil fields," Natur og Ungdom leader Gaute Eiterjord said in a statement.
"The Norwegian oil policy is betraying my generation and threatening our future, which is why we are appealing," he added.
The environmentalists accuse Norway of violating an amended article of the country's constitution that guarantees since 2014 the right to a healthy environment.
But the Oslo court said the state could not be held responsible for CO2 emissions caused by hydrocarbons which it exports to other countries.
"There is already enough carbon dioxide in the atmosphere to seriously damage our future," Truls Gulowsen, head of Greenpeace in Norway, said in a statement.
"By opening up these pristine areas for oil exploration Norway is effectively smuggling its emissions outside of its own borders and furthering climate change, which harms everyone, everywhere," he added.
Greenpeace raised 300,000 kroner (31,000 euros) in donations after the Oslo court ordered the NGOs to pay for the state's legal costs of 580,000 kroner, Gulowsen told AFP.
The state's defense lawyer said he could not comment on the decision as he had not yet received the documents.
The appeal signals that the fight against global warming is increasingly brought to courtrooms.
The NGOs argue that new oil activities in the fragile Arctic region would be contrary to the 2015 Paris climate accord, which seeks to limit average global warming to under two degrees Celsius.
As oil production has been halved since 2000, Norway has turned its sights toward the far North: the Barents Sea holds 65 percent of the undiscovered reserves on the Norwegian continental shelf, according to the Norwegian Petroleum Directorate.
Latest drilling campaigns, however, have yielded disappointing results.
The Norwegian state in May 2016 granted exploration licenses in the Barents Sea to 13 companies, including it's national champion Statoil, US giants Chevron and ConocoPhillips, and Russia's Lukoil.
If the Supreme Court of Norway refuses to hear the case, then it will be sent to a lower appeals court.