The attack on Tuesday morning, the biggest targeting Moscow in months, came as a Ukrainian delegation was set to meet with America's top diplomat in Saudi Arabia about ending the three-year war with Russia.
There was no immediate comment from Ukrainian officials on the attack.
At least 11 of the drones were shot down in the Ramensky and Domodedovo districts of the Moscow region surrounding the Russian capital, Sobyanin said on the messaging app Telegram.
He did not specify where the other drones were shot down, noting only that they were "flying towards Moscow".
The governor of the Moscow region, Andrei Vorobyov, said that one person was killed and three more wounded as a result of the drone attack.
The attack damaged seven apartments in a residential building in the Ramensky district, he said.
Sobyanin said the roof of a building in Moscow also sustained damage, which he described as "insignificant".
Video of the building, published by Russia's state news agency RIA Novosti, showed a charred spot on the facade of a multi-storey residential building near the roof, with bits of the building's lining stripped off.
Flights had been restricted in and out of four airports, including Domodedovo, Vnukovo and Zhukovsky to the south of Moscow and Sheremetyevo to the north, according to Russian civil aviation authority Rosaviatsia.
Train traffic through the Domodedovo railway station had also been briefly halted, local officials reported.
Authorities also reported shooting down or jamming drones in the Kaluga, Ryazan, Tula and Vladimir regions next to the Moscow region, as well as the Belgorod region on the border with Ukraine.
Rosaviatsia also restricted flights in and out of airports in the Yaroslavl and Nizhny Novgorod regions.
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