The six protesters used ropes and harnesses Monday to scale the huge platform in the Pacific Ocean, tweeting images of their daunting climb as they went.
"We made it! We're on Shell's platform. And we're not alone," wrote Aliyah Field, an American activist taking part in the protest. "Everyone can help turn this into a platform for people power!"
The rig, the Polar Pioneer, is on its way to the Arctic via Seattle. The environmental activists caught up with it about 750 miles northwest of Hawaii, Greenpeace said
They plan to occupy the underside of the rig's main deck and say they have enough supplies to stay there for several days.
Shell didn't immediately respond to CNN's request for comment late Monday. But company spokeswoman Kelly Op De Weegh told Agence France-Presse that the boarding was illegal and jeopardized the safety of the activists and the crew.
Greenpeace is furious over a decision last week
by U.S. authorities to lift the suspensions on leases to drill for oil and gas in the Chukchi Sea, which lies between northern Alaska and Russia. Shell and several other oil companies bought exploration leases for the sea in 2008.
The U.S. government's decision "means that in 100 days, Shell could begin drilling in the Alaskan Arctic," Greenpeace said.
Federal agencies still have to review and approve companies' plans before exploratory drilling can start.
Shell's work in the Arctic
has suffered some setbacks
in the past, including a drilling barge that ran aground
off southern Alaska on December 31, 2012.
Greenpeace has repeatedly used the occupation of drilling-related vessels to bring attention to its cause. One group of its activists was detained for months
by Russian authorities in 2013.
In the current episode, the activists pursued the Polar Pioneer, which Shell is leasing from Transocean, as it traveled thousands of miles aboard a transport vessel from Malaysia, Greenpeace said.
Besides the American protester Field, the activists come from Australia, Austria, Germany, New Zealand and Sweden.
The U.S. Coast Guard wasn't immediately available for comment late Monday on the incident.