The girl, called Nieve, was attending the Handmade Parade in Hebden Bridge on Sunday with her grandparents when the dog jumped up and bit her.
She needed eyelid surgery and suffered "potentially irreparable damage" to a tear duct.
The man in the e-fit is described as being broad, bald and aged about 50.
A second man, also thought to be about 50 and of similar build, was present at the time of the incident, West Yorkshire Police said.
Both men left the scene without checking on the girl's welfare or leaving any details.
The dog, believed to be a Staffordshire bull terrier, was on a lead at the time of the incident and is described as brindle and white coloured.
PCSO Gareth Helliwell said: "The five-year-old girl involved in this incident needed reconstructive surgery to an eyelid following this attack and has suffered potentially irreparable damage to her lower tear duct."
The girl's family described the dog's owner as "scum", adding "she could have lost her eye, she was covered in blood and the guy just ran off, it's cowardly".
She was also bitten on the ankle despite the dog being pulled away on a lead, they said.
Police appealed for anyone with information about the dog or the two men with it to contact them.
Police have issued an e-fit of a man they want to trace in connection with a dog attack on a five-year-old girl.