Reports say that electrical cables were damaged as Moscow was lashed with high winds, hail and torrential rain.
The exact death toll is unclear, but Russian news agencies are reporting between six and 11 fatalities.
The city's investigative committee said that "hurricane winds" had caused trees to fall in various parts of the city, killing five pedestrians.
An elderly man was also killed at a bus stop, it said. TASS news agency says that 69 people have been injured.
Two people were killed after a tree fell onto their summer house, Interfax reported.
BBC Weather: Forecast for Moscow
Delays had been reported on some metro lines but normal service appears to have resumed.
Mr Sobyanin tweeted: "My condolences to the friends and family of those killed."
Photos circulating on social media show fallen trees blocking roads and a large roadside advertising board crushing parked cars.
A video also shows strong winds tearing apart the roof of a covered tennis centre in Zhulebino district.
Russian President Vladimir Putin is in Versailles, near Paris, where he has been holding talks with French President Emmanuel Macron.
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