The 29-year-old reporter was shot by husband Anatoly Leontikov, it has been reported.
Russian media outlets also reported Veber was shot in front of her nine-year-old son Dmitry.
According to police, Leontikov fled with Dmitry, but later turned himself in.
Veber split from Leontikov two years ago but the pair were not officially divorced, local media reports.
Veber was an employee at Laba TV and the station's official statement described her as a 'talented journalist.'
Fans of the reporter have flooded her social media with touching tributes.
Our thoughts go out to her family at this difficult time, especially her son Dmitry.