Footage posted on social media showed smoke billowing over the city as a result of a large blaze, which grounded all flights to the airport. No casualties have been reported.
Russia’s defence ministry also claimed that more drones were shot down over several other cities, including Oryol, Bryansk, Ryazan, Kaluga and the Moscow region.
Some of the drones were said to have hit an empty oil reservoir, while another was said to have been heading for a TV tower.
Moscow’s three main airports, Sheremetyevo, Vnukovo and Domodedovo, temporarily halted flights, a measure that has become routine.
Russian state media claimed between ten and 20 drones were involved in the attack on Pskov, the only region to report extensive damage.
Separately, in Kyiv, falling debris killed two people and injured another after Russia launched a “massive combined attack” using drones and missiles, head of the military administration Sergei Popko wrote on Telegram.
In what Mr Popko said was the biggest attack since spring, Russia launched Shahed drones at Kyiv from various directions, with 20 targets brought down by Ukranian air defence.
The deaths and injury occurred when debris fell on a commercial building in the Shevchenkivskyi district, Mr Popko said.
Explosions in Ukraine were also reported in the southern city of Odesa and the Cherkasy region.
Ukraine usually does not publicly take responsibility for drone attacks on Russian soil, although President Volodymyr Zelensky has previously described them as a “fair” response to Putin’s unprovoked invasion of Ukraine.