The two men, calling themselves ‘UrbanMonkeys', then circulated a photo and a video of themselves on Facebook climbing the stack.
One man was wearing a mask but the second will be clearly recognisable to anyone who knows him.
The daredevils are ‘roofers' - people who make it a hobby to climb to the top of skyscrapers, bridges, chimneys and other man-made constructions, and then take photos or video to document their achievements. They climb in normal trainers and without a rope or harness.
Roofers rarely seek permission, and run the risk of being arrested. The stunts are particularly popular among teenagers in eastern Europe.
The disused nuclear plant is owned by Austrian energy company EVN and serves as a training facility for international nuclear engineers and occasionally as a film set.
"We're always open to quirky and interesting ideas at Zwentendorf - but please make an appointment. As stunning and cool as the photos are - we do have to notify the police for something like this,” EVN's Stefan Zach said in a press release.
He added that the group had caused “some significant damage” by breaking into the plant - including to the gates, the site's video surveillance system and the emergency lighting system in the chimney.
Writing on their Facebook page on Wednesday, the UrbanMonkeys said that "climbing is not a crime" and denied that they had committed vandalism. "With such weak security, there was no reason to damage the gates or the cameras," they added, complaining that the media were attempting to portray them as criminals.
Zwentendorf was meant to be the first of several planned nuclear plants destined to supply electricity for Austria but after a referendum in November 1978, the one-billion-euro project stayed offline and Austria has since remained firmly anti-nuclear.