Ghoncheh Ghavami, a 26-year-old University of London graduate, is still subject to a travel ban, however, and cannot leave Iran, the family was quoted as saying.
"She is free," said her brother, Iman Ghavami.
"The court of appeal has waived the remaining seven months of her sentence and she will not have to go back to jail," he told The Guardian.
Ghavami's case attracted huge international attention and the "Free Ghoncheh Ghavami" campaign included a Facebook group and a Change.Org online petition.
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"My mum and I will not forget your generous support and thank you sincerely. Together we brought Ghoncheh home," he said.
Ghavami was detained outside Azadi ("Freedom" in Persian) Stadium in the capital on June 20, having gone there to watch a volleyball match.
She was refused entry as women are not allowed to watch male athletes in the Islamic republic.
Officials have said that Ghavami was arrested for security reasons unrelated to the sporting event.
In November, a justice official said she was arrested for having contacts with the "opposition based abroad" and propaganda offences against the regime.
Iran has since broken with custom and will allow foreign women to attend male matches in a separate part of the stadium at a major tournament this year.