Three senior judges said that it was not acceptable for the mother to block the father's reasonable efforts to see his two daughters.
They made their judgment at the end of a legal battle between the unmarried couple who'd had a six-year relationship.
In 2008, the mother left home without warning, taking her daughters, who were then aged four and one, and refusing to let the father see them.
The Daily Telegraph reported that in February this year, the father, from the Swindon area, was banned by a family judge from having any direct contact with his children, now nine and six.
The father was told he could send his daughters cards, letters and gifts once a month, but could not see them because it would be too distressing for their mother.
The county court judge said that she would be 'unable to cope' with the father seeing the children after she broke down in court and said the thought of it made her feel 'exhausted'.
The father's lawyers challenged the judge's decision, saying it had been based on a 'momentary' display of emotion from the mother in the witness box and the views of a 'lone' psychologist, who supported her case.
The decision also went against the view of the independent advocate appointed for the children.
The father's barrister, Sarah Evans, told the Court of Appeal: "It is a fundamental tenet of the law of this country that, save in exceptional circumstances, children have a right to a meaningful relationship with both parents."
The appeal judges accepted that the father's exile from his own children's lives violated his rights. Lord Justice McFarlane recognised that it was 'a very big ask' for the mother to accept that her children's best interests lay in having two parents, not just one.
Where, however, it is plainly in the best interests of a child to spend time with the other parent then, tough or not, part of the responsibility of the parent with care must be the duty and responsibility to deliver what the child needs, hard though that may be," he said.
The process will take place under the supervision of a court-appointed guardian.
The judges urged all separated parents to see the 'bigger picture' and consider the harm that legal disputes cause children.