"We have offered our sincerest apologies to these patients and their families," University Hospitals said in the statement.
"We recognise they entrusted us with their care. The situation is entirely inconsistent with our commitment to helping patients return to health and live life to the fullest."
University Hospitals officials added that investigations into the incident from both "outside agencies" and "internal quality and compliance experts" are ongoing.
"UH will take the actions necessary to prevent such a mistake from ever happening again," University Hospitals said.
The health care system also notified the United Network for Organ Sharing, the organisation that manages the transplant system, of the mistaken surgery.