Ms Tsurkov entered Iraq with her Russian passport, according to Iraqi authorities.
She was working on research for her doctoral degree at Princeton University in the US but had not been seen since March after a suspected kidnapping.
“It is encouraging to see this proof of life video, that was clearly filmed in recent weeks,” the family said.
“While we can’t comment on what Elizabeth is saying, we appreciate that this is an important step in the process to bring her home to her family,” a statement added.
Sitting in a room and wearing a black shirt Ms Turkov, 37, appealed for her release and an end to the Israel-Gaza war in a 4-minute video.
“I have been in captivity for seven months now,” she said. “There is nothing to secure my release, I’m in a difficult situation.
“To my family … work to win my release as soon as possible in order to return to you,” she said.
In the video, Ms Tsurkov said that she had worked as an agent for Israel's Mossad and the CIA in Iraq and Syria, an accusation that social media accounts linked to Iran-backed militias in Iraq circulated after her disappearance. They have offered no evidence for the claim.
While in Syria, she said her role was to establish a connection between Israel and the US-backed Syrian Democratic Forces.
In Iraq, she worked to co-ordinate pro-democracy 2019 protests that swept through Baghdad and provinces in the south of the country.
“We worked to deepen the differences in order to create a Shiite-Shiite warfare inside Iraq,” she said.
She also commented on the Israel-Gaza war that began early last month and called for it to end, describing the Israeli government policy as “foolish”. She said it would “lead to hatred between us and the Palestinian people in Gaza and will not achieve peace”.
The National could not independently verify the authenticity of the video. Videos of prisoners meant to document a proof-of-life often include statements coerced by captors or statements otherwise made under duress.
The Iraqi Interior Ministry is collecting data and investigating the video, an official said.
No group has claimed responsibility for the kidnapping, but Israeli Prime Minister Benjamin Netanyahu said earlier this year that she was being held by the powerful Iran-backed Iraqi militia Kataib Hezbollah.
The group has denied the accusation.