The Russian defence ministry said two drones were "suppressed and crashed", adding that there were no casualties.
Russia's state-owned Tass news agency reported that one drone fell close to the defence ministry.
Ukrainian officials are yet to comment, but they rarely claim responsibility for attacks inside Russia.
In a separate development, Russian-installed officials ordered people to evacuate one district of Crimea - the Ukrainian southern peninsular annexed by Moscow in 2014 - after a reported overnight Ukrainian drone attack.
Kremlin-appointed regional head Sergei Aksenov said an ammunition depot was hit in the northern Dzhankoi area, and residents of nearby villages were told to leave their homes. He reported no casualties.
Meanwhile, Ukraine said four people were injured and a grain hangar was destroyed in new Russian drone attacks on its southern Odesa region. It said the country's port infrastructure on the Danube was targeted.
Russia has been launching near-constant attacks on the Odesa region - where Ukraine's major Black Sea ports are located - since Russia withdrew from a landmark grain deal last week.
In Moscow, the city's Mayor Sergei Sobyanin said the drone strikes in the Russian capital hit "non-residential" buildings around 04:00 local time (01:00 GMT). In a post on social media, he added that the buildings had not sustained any major damage.
But state news agencies reported that some drone fragments were found just 2km (1.2 miles) away from the defence ministry's buildings.
"A Kyiv regime attempt to carry out a terrorist act using two drones on objects on the territory of the city of Moscow was stopped," the Russian defence ministry said in a statement.
Russian news agencies reported that drone debris was also found along the city's Komsomolsky Avenue. Moscow's transport department wrote on Telegram that traffic along the route was blocked and photos showed emergency services working at the scene.
Traffic was also stopped on Likhachev Avenue, where a high-rise office building was damaged. Footage published on the military Zvezda TV channel showed missing windows at the top of the building.
Moscow has accused Ukraine of a host of drone attacks on its territory in recent months.
Earlier this month, Russia said Ukraine launched a drone attack on Moscow, forcing flights to be diverted from Vnukovo International Airport. Ukraine did not claim responsibility.
And in May Kyiv denied carrying out a drone attack on the Kremlin, which Russia said amounted to an attempt on President Vladimir Putin's life.
Monday's attack comes just a day after Russia targeted Odesa with missile strikes, destroying the historical Transfiguration Cathedral.
Unesco, the UN's cultural agency, said it was "deeply dismayed and condemns in the strongest terms" the attack on the historic centre of Odesa.
Ukraine's President Volodymyr Zelensky vowed retaliation for the attack.
"They will definitely feel this," he said during his nightly address from Kyiv. "The target of all these missiles is not just cities, villages or people. Their target is humanity and the foundations of our entire European culture."