The three-month-old was born with a diaphragmatic hernia, a rare, life-threatening condition which meant she had a hole in her diaphragm.
All the organs in her stomach were pushed into her chest, squashing her lungs and preventing them from developing properly.
As soon as she was born she was given emergency CPR to keep her alive and rushed to intensive care.
At one point doctors told mum Tahira: "Tell all your family to come in so they can see her once."
Hawwa, which means Eve in Arabic, is now making excellent progress after treatment on ECMO - or extracorporeal membrane oxygenation - which does the work of the heart and lungs allowing surgeons to carry out life-saving surgery.
The national service, based at Yorkhill hospital, has just been presented with the coveted ELSO Award for Excellence in Life Support.
Tahira, 25, a student nurse from Crosshill, who is married to Waseem, 26, was given the shock diagnosis when she was 17 weeks pregnant.
Yorkhill hospital will see only around ten babies each year with the condition.
When she was born at the city's Southern General, a team of medics took her away for emergency care. Tahira didn't know if she had given birth to a boy or a girl.
Hawwa was too unwell to be moved to Yorkhill so the equipment had to be taken there.
Tahira, who has another daughter Imaan, 18 months, said: "It was a shock. She was born at 3pm but I didn't see her for hours.
"Even when she went on to ECMO the doctors didn't know if she would survive.
"She looked so tiny and fragile when she was on the machine but it has worked. Hawwa is still in hospital but we hope to have her home with us very soon."
When she was a week old, Hawwa had surgery to move her stomach organs into the proper position and repair the hole in her diaphragm and doctors are hopeful she will lead a normal, healthy life.
Gillian Wylie, who coordinates the national service, said: "Happily for us most babies that go on to ECMO do really well."
The life-saving service has treated more than 500 paediatric patients from across Scotland and further afield since it began in 1992.