It marks the first time such a fossil has been discovered in Scotland outside of Skye.
The limb bone is thought to belong to a stegosaurian dinosaur, such as a stegosaurus.
Dr Elsa Panciroli, research affiliate at National Museums Scotland, stumbled across the fossil in what she said was a “serendipitous discovery”.
“I was running along the shore on my way back to meet the rest of the team and I ran right over it,” she said.
“It wasn’t clear exactly what kind of animal it belonged to at the time, but there was no doubt it was a dinosaur bone.”
She said that in 200 years of searching the area “no-one has found a dinosaur before, so this is quite special”.
She added: “This is a hugely significant find.
“Globally, Middle Jurassic fossils are rare and until now the only dinosaur fossils found in Scotland were on the Isle of Skye.
“This bone is 166 million years old and provides us with evidence that stegosaurs were living in Scotland at this time.”
The bone dates to the Middle Jurassic period and is just over half a metre long.
It was found in a boulder on the foreshore.
Though it was badly damaged by waves, there was enough remaining for a team of palaeontologists to study.
The bone was extracted and brought to a laboratory to be removed from the rock where it was found to be part of the hind limb of a stegosaur.
It dates to the same period as similar fossils found on Skye.
The Isle of Eigg is already known for its Jurassic fossils, particularly marine reptiles and fish, first discovered by 19th century geologist Hugh Miller.
Dr Steve Brusatte of the University of Edinburgh co-authored a paper on the find.
He said: “Elsa’s discovery of this bone is really remarkable.
“Nobody, not even Hugh Miller himself, had found dinosaur bones on Eigg before.
“This fossil is additional evidence that plate-backed stegosaurs used to roam Scotland, which corroborates footprints from the Isle of Skye that we identified as being made by a stegosaur.”
The bone is now in the collections of National Museums Scotland in Edinburgh.
The fieldwork on the Isle of Eigg was funded by the National Geographic Society with the permission of The Isle of Eigg Heritage Trust.
The full paper is published in Earth And Environmental Transactions Of The Royal Society Of Edinburgh and is available at https://cambridge.org/eiggdinosaur