The RSPCA has launched an investigation following the 'disgusting' incident on July 26.
The cygnet died shortly after it was attacked.
Witnesses spotted two men laughing as their dog, an akita-type breed, attacked the cygnet on Wednesbury Oak Loop Canal Path.
The path leads between Ladymoor Road Bridge and Highfields Road Bridge, in Bradley, near Coseley.
The charity has now released a video of the men and their dog, in a bid to track them down.
WATCH: Do you know these men?
RSPCA Inspector Nayman Dunderdale said: "The dog grabbed hold of the cygnet and dragged it down the path for around 30m while they laughed.
“A member of the public shouted at them and the dog threw the bird into the air before letting it go. They then reported that the men went over to pick it up and tried to break its neck before walking off.”
Advisory: Distressing images
The cygnet died a short while after the attack. Two adult swans and four other cygnets remained on the water. Police have been informed and the men were described as Asian.
Inspector Dunderdale added: “As a dog owner you should always have your pet under control in public spaces - particularly when there are people or animals around.
“We encourage owners to keep their dogs on leads when they’re in areas with wildlife and wild birds.
“This is an awful incident which left a poor, defenceless cygnet with devastating injuries that ultimately cost it its life.
“It’s disgusting that these men allowed their dog to attack this vulnerable bird in such a violent manner and that they didn’t attempt to intervene and help the bird. In fact, they seemed to find the whole event entertaining which is simply despicable."
Anyone with information is urged to call the RSPCA appeal line on 0300 123 8018.