The activists "have taken up positions" on the platform and have enough supplies for "several days", Greenpeace said.
The group said three inflatable speedboats had approached the platform at the Prirazlomnoye oil field after 4 am to let six people, including the organisation's global chief Kumi Naidoo, climb it via mooring lines using ropes and ladders.
The Prirazlomnoye field, Russia's first Arctic offshore development, has been plagued by delays due to cost overruns and platform construction difficulties, with oil now expected to flow at the turn of the year.
"The only way to prevent a catastrophic oil spill from happening in this unique environment is to permanently ban all drilling now," Naidoo said.
A Greenpeace video of the boarding
An eyewitness aboard a Greenpeace ship some 3.5 miles away from the platform said activists have taken portable tents, along with the supplies.
Gazprom and Gazprom Neft Shelf, a subsidiary holding the Prirazlomnoye license, declined immediate comment.
Prirazlomnoye is estimated to hold reserves of 526m barrels and success in launching Arctic oil exploration is seen vital to sustaining Russia's long-term status as the world's top oil producer.
But environment campaigners warn Arctic's extreme conditions - remoteness, fragile ecosystems, darkness, sub-zero temperatures, ice, or high winds - are likely to stall any emergency operations in case of a spill.
Greenpeace will seek to promote a resolution at the United Nation's general assembly that would declare the Arctic part of the global commons, Naidoo said this month in Moscow, adding that would prevent any drilling there.