Four-year-old Mylee Weetman, from Doncaster, died in 2013 after surgery at the children's cardiac unit at Leeds General Infirmary (LGI).
Mylee had been diagnosed with a congenital heart defect and was starved of oxygen during surgery, the inquest in Wakefield had heard.
Coroner David Hinchliff recorded a narrative verdict.
Her mother Siobhan Casey had told the inquest no-one had been able to explain to her how or why she had died.
Mylee was diagnosed with the congenital heart defect Tetralogy of Fallot after her birth in 2009. She required an operation in 2010.
A further operation to repair damage to her heart took place at the LGI on 15 March 2013 during which she suffered an embolism, a "rare but recognised complication of this necessary surgery" according to the coroner's verdict.
Mylee died in the early hours of 21 March.
In a statement after the verdict the girl's family thanked the coroner for a "painstaking" investigation and said it had been "so distressing to relive the events around Mylee's death".
"We still, however, have many questions about Mylee's treatment at Leeds," it said.
The family expressed the hope that lessons would be learned.
Dr Yvette Oade, for Leeds Teaching Hospitals NHS Trust, reiterated her "deepest sympathy" to Mylee's family.
"I hope this hearing, although painful for them, will have provided the answers they needed," she said.
"The coroner concluded that while the surgery was performed correctly, Mylee sadly suffered a micro-embolic air embolism which is a rare but recognised complication of this procedure."
Mylee's death came amid concerns about death rates at the unit in Leeds. Surgery was later suspended for two weeks, but a review found the centre was safe.
A child died after a "rare but recognised complication" of heart surgery, a coroner has ruled.