Sarah's surgery lasted about 6 hours and doctors had no major challenges in resizing the adult lungs to fit into her 10-year-old chest, CNN reported.
She is still intubated and will remain so for another 48 hours.
"Sarah is in the process of getting settled in the ICU, and now her recovery begins," the family told CNN in a statement. "It will be a long road, but we're not going for easy, we're going for possible. And an organ donor has made this possible for her."
Infection and organ rejection are still risks.
Her mother, Janet, earlier announced the impending operation on Facebook in a post that began "God is great." In a later post, Murnaghan said Sarah was in an operating room at Children's Hospital of Philadelphia.
"Please pray for Sarah's donor, her HERO, who has given her the gift of life," Murnaghan said. "Today their family has experienced a tremendous loss, may God grant them a peace that surpasses understanding. Please pray for Sarah and her surgical team and our whole family!"
The 10-year-old Newtown Square, Pa., girl has cystic fibrosis and her parents said she had only a few weeks to live without a lung transplant. She was on a list for a pediatric transplant, but last week a federal judge in Philadelphia ordered Sarah and another patient at the same hospital, Javier Acosta, 11, of New York, moved to the adult list pending a full court hearing.
The Organ Procurement and Transplantation Network, which supervises organ donations in the United States, Monday changed its rules to allow children less than 12 years of age to be considered for the adult list.