Zookeepers realised the tiny 73g chick, called Rainbow, was still alive and gently removed her from the shell before moving her into an incubation room.
Rainbow was given a cuddly toy penguin in place of her mother and kept under the warmth of a heat lamp.
She was also hand-fed three times a day with a “penguin milkshake”, consisting of blended fish, vitamins and minerals.
The one-month-old chick will stay in the incubation room until she reaches 10 weeks old and around 3kg in weight.
Penguin keeper Suzi Hyde said: “We were overjoyed when she started begging for food by opening her mouth wide and making tiny squawks.
“It was the first sign that she might just make it.”
She added: “Penguins do accidentally step on their eggs, which – even if the chick survives – invariably leads to them rejecting the infant.”
Rainbow is expected to move to the zoo’s penguin nursery, where she can learn to swim.
She will then join the 80 other penguins at their 1,500sq m pool, which is the biggest in Britain.