The woman's QC, Jenni Richards, told the three judges that she wants to fulfil her daughter's wishes that her mother should carry a child created from her frozen eggs "and raise that child".
The woman and her 59-year-old husband are challenging the decision of Mr Justice Ouseley last June to dismiss their case.
The judge heard that the daughter, who can only be referred to as "A" for legal reasons, was desperate to have children and asked her mother to "carry my babies".
Her parents launched legal action against an independent regulator's refusal to allow them to take their daughter's eggs to a US fertility treatment clinic to be used with donor sperm.
The Human Fertilisation and Embryology Authority (HFEA) said the eggs could not be released from storage in London because A did not give her full written consent before she died of bowel cancer at the age of 28.
Giving the court's ruling, Lady Justice Arden said the challenge succeeded at three levels.
"First, there was on the face of it the misstatement of certain of the evidence about A's consent by the committee.
"Second, even if what the committee meant was that there was a lack of effective consent because the appellants could not show that A received information on certain matters, the decision was flawed because the committee pointed to the lack of certain evidence without explaining why A needed to receive that information and give that consent.
"The third level is that the committee did not ask the prior question of what information the Human Fertilisation and Embryology Act required to be given to A in the circumstances of her case."
She said the decision must be set aside and remitted to the committee for further consideration of Mr and Mrs M's export application.
In the High Court, Mr Justice Ouseley ruled that the HFEA had been entitled to find the daughter had not given "the required consent". He declared that there had been no breach of the family's human rights.
Ms Richards argued at the Court of Appeal that there was "clear evidence" of what A wanted to happen to her eggs after she died, and that "all available evidence" showed she wanted her mother "to have her child after death".
The appeal is opposed by the HFEA, which says that Mr Justice Ouseley "did not err in concluding that the HFEA's decision was lawful".
Catherine Callaghan, counsel for the HFEA, said in written argument before the court: "It is natural to feel sympathy for the appellants' loss and for their wish to keep their daughter's memory alive by trying to conceive a child using their daughter's eggs.
"However, the court's role is not to decide whether it would have permitted the mother to undergo fertility treatment using her deceased daughter's eggs and donated sperm.
"Rather, its role is to determine whether Mr Justice Ouseley erred in concluding that the HFEA's statutory approvals committee acted lawfully and rationally in exercising its broad discretion to refuse to authorise export of the frozen eggs to a treatment centre in New York for use in the way proposed."
