The discovery was made by researchers studying a rare, mummified specimen of Edmontosauraus regalis.
The most common dinosaurs in North America between 75 and 65 million years ago, duck-billed dinosaurs were gentle giants, about 12 meters long.
It was never suspected that they - or other dinosaurs, for that matter - had fleshy structures on the tops of their heads.
"Until now, there has been no evidence for bizarre soft-tissue display structures among dinosaurs; these findings dramatically alter our perception of the appearance and behaviour of this well-known dinosaur and allow us to comment on the evolution of head crests in this group," said Phil Bell from Australia's University of New England.
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The dinosaur specimen that the researchers studied was found in deposits west of the city of Grande Prairie in west-central Alberta, Canada.
Bell, along with Federico Fanti from the University of Bologna, Italy, knew they had something special when they found skin impressions on parts of the mummified body.
But it wasn't until Bell put a chisel through the top of the crest that he realised they really had something incredible.
"An elephant's trunk or a rooster's crest might never fossilise because there's no bone in them," Bell said.
"There's no reason that other strange fleshy structures couldn't have been present on a whole range of other dinosaurs, including T rex or Triceratops," Bell added.
It is not yet known what cocks comb might have done for the duck-billed dinosaurs. In roosters and some other birds, bright red crests are a way to attract mates.
"We might imagine a pair of male Edmontosaurus sizing each other up, bellowing, and showing off their head gear to see who was the dominant male and who is in charge of the herd," Bell said.