"But she has still got a long way to go before she is out of the woods.
Thirteen-month-old Margaux Bride, from Tooting, south London, underwent the live-saving operation on Friday night after a donor organ became available on the continent.
Doctors from the Royal Brompton Hospital in London flew out to collect the organ having only a four hour window to perform the transplant.
Margaux, who was dying of multiple heart defects, is in a critical condition.
She had undergone 14 previous operations, including five which involved open-heart surgery.
Surgeons pleased
A hospital spokeswoman said: "The baby has undergone major heart surgery which the surgeons are very pleased with.
"But she has still got a long way to go before she is out of the woods.
"Every day is critical, she is fighting hard and she is doing well."
Maugaux's parents, Claude and Martin, said they were overjoyed when they were told an organ had been found for their daughter.
Mr and Mrs Bride had already made a desperate television appeal for a heart donor last month after doctors told them a transplant was her only hope of survival.
Their daughter was resuscitated twice after suffering cardiac arrests on Wednesday and Thursday and the couple had been told she was terminally ill.
The couple were told on Friday lunchtime that a suitable heart could be available.
Mrs Bride said: "She is very well. She has opened her eyes several times and she has held my hand - I could not believe it.
"The weekend has been extremely draining and worrying. Just when we were about to give up hope they called us saying a heart was available."