Zann was born in Sydney, Australia, the daughter of Janice, a high school teacher, and Paul Zann, a professor. Her great-grandfather, named Zanninovich, came to Australia from Yugoslavia.
Zann emigrated with her parents to Canada in 1968, first to Regina, Saskatchewan, then to Truro, Nova Scotia, and later graduated from Cobequid Educational Centre, a high school in Truro which was noted for its student musical productions. She graduated from York University in Toronto where she studied drama, fine arts and political science.
Zann worked as a screen, television, stage, and voice actress and appeared in numerous television shows and films. As an actress, she is best known for providing the voice of Rogue on the 1992 animated television adaptation of the X-Men comic book series.
After living in Halifax, Toronto, London, Stockholm, Vancouver, Los Angeles, and New York City, with her dog Buster Keaton, in 2008 Zann moved back to her adopted hometown of Truro while still retaining apartments in New York City, Vancouver, and Toronto. That fall she started a community campaign to restore a historic former academic building in downtown Truro, the Provincial Normal College, into a cultural centre.
On 8 April 2009, Zann successfully ran for the Nova Scotia New Democratic Party nomination in the riding of Truro-Bible Hill. On the same day, an employee of the Liberal Party leaked a topless photo of her to the media from her appearance in The L Word. Zann was elected in the 2009 provincial election and served as the Ministerial Assistant for the Department of Tourism, Culture, and Heritage. She was subsequently re-elected in the 2013 provincial election to represent the reconfigured electoral district, now called Truro-Bible Hill-Millbrook-Salmon River.
In that election the NDP was reduced to seven seats in the Legislature and no longer formed the government. Currently Zann is party critic for Aboriginal Affairs, Community Culture and Heritage, Community Services, Education, Human Rights Commission/Status of Women, and Seniors and Disabled.
Zann has also taken a lead position in regard to environmental racism and has introduced Bill-111, the Environmental Racism Prevention Act, to address this issue in Nova Scotia.
On June 12, 2015, Zann announced her bid for the leadership of the Nova Scotia New Democratic Party. On February 27, 2016, Zann was defeated in her leadership bid, losing to Gary Burrill on the second ballot.