"Yevgeny tried to grab Alexander and jumped after him, but unfortunately, they both died after falling from a high cliff into a mountainous river," it said.
Margarita Simonyan, editor-in-chief of Russia's RT television channel, commented separately on Twitter.
"There were a fair number of witnesses - no one even had time to think about what had happened when Zinichev threw himself into the water for the fallen person and crashed into a protruding rock," she said.
The ministry had said in a statement earlier on Wednesday that Zinichev visited the construction site of a new fire station in Norilsk, as well as a search and rescue team in the region.
A biography carried by state-run media said Zinichev was born in St Petersburg and served in the state security services from 1987 to 2015 and was also involved in security for Putin.
He served as a bodyguard and personal aide to Putin, according to the RBC news site. Multiple photographs of Putin at public events dating over several years show Zinichev standing at Putin's shoulder.
Zinichev later became deputy head of the federal security service and served briefly as acting regional governor of Kaliningrad. The Kremlin said Putin had offered his condolences.