The strike involved eight drones that were repelled “satisfactorily”, Mr Putin said in televised comments on Tuesday.
“It’s clear what needs to be done to seal the air defence in the capital,” he said.
The Russian Ministry of Defence blamed Ukraine for the assault, saying missiles shot down five of the drones while electronic jamming was used to divert three others from their intended targets.
The Ministry of Foreign Affairs threatened “the harshest” retaliation against Kyiv for the attack.
The drone strike was what one politician called the “worst attack on the capital since World War Two”.
Russia launched its own attack overnight, killing four people across Ukraine.
There was no immediate claim of responsibility for the attack on the Russian capital.
Ukraine is poised to launch a counter-offensive backed by western weapons to try to drive Russian occupiers out of territory seized since Moscow launched its “special military operation” in February last year.
Speaking at Bratislava’s GlobSec conference, Ukraine's State Secretary for Defence, Kostiantyn Vashchenko, said: “Now we have finished all preparation and we are really ready to go ahead.”
Ukrainian presidential aide Mykhailo Podolyak denied Kyiv was directly involved in attacking Moscow, although he said “we are pleased to watch” and forecast more to come.
White House spokeswoman Karine Jean-Pierre said the US did not support attacks inside Russia.
We “are certainly gathering information about what happened to get a better perspective and some clarity about what happened”, Ms Jean-Pierre told reporters.
“As a general matter, we have said this before, we do not support attacks inside of Russia. We've been very clear about that.”
The early morning raid was aimed at some of Moscow's wealthiest areas, including a western zone where Mr Putin and other elites have homes.
Two people were injured while some Russians in two lightly damaged apartment blocks were briefly moved to safety, said Moscow's Mayor, Sergey Sobyanin.
Residents said they heard loud bangs followed by the smell of petrol. Some filmed a drone being shot down and a plume of smoke rising over the Moscow skyline.
Russian politician Maxim Ivanov called it the most serious assault on Moscow since the Nazis, saying no citizen could now avoid “the new reality”.
“You will either defeat the enemy as a single fist with our motherland, or the indelible shame of cowardice, collaboration and betrayal will engulf your family,” Mr Ivanov said.
Andrey Vorobyov, Governor of the wider Moscow region, later said several drones were “shot down on the approach to Moscow”.
The incident comes weeks after two drones exploded over the Kremlin – an attack Russia blamed on Ukraine, although Kyiv denied responsibility.
It follows intense Russian drone and missile strikes on Ukrainian cities this month.
Moscow targeted by drones – in pictures
In Kyiv, more than 20 drones were shot down during a wave of strikes that followed a barrage of missiles in a rare daylight attack on Ukraine’s capital city on Monday.
Four people died across Ukraine, including one in the capital, due to the attacks, which wounded 34, including two children.
In Kyiv, the death resulted from debris from a destroyed Russian missile, which hit a high-rise apartment building causing a fire, officials said. Four people were injured.
Two upper floors were destroyed with people possibly still under rubble. Flames engulfed the top of the building.
Kyiv's military administration said that, unlike most previous raids, Tuesday’s strikes involved only Iranian-made Shahed drones and no missiles.
“Russia is trying to break us and break our will. To deprive peaceful citizens of sleep and prevent us, the security and defence sector, from preparing to implement important tasks,” said Ukraine's Interior Minister Ihor Klymenko.
Debris hit several other parts the capital including the historic Podil and Pecherskyi neighbourhoods.
It was not immediately known how many drones Russia launched and there was no immediate comment from Moscow.
Russia has repeatedly attacked Kyiv in May using drones and missiles, mostly at night, in an apparent attempt to undermine Ukrainians' will to fight after more than 15 months of war.
Drone attacks in Kyiv – in pictures
Tuesday's strikes were Russia's 17th air assault on the capital this month and came after the city was attacked twice on Monday.
“These missile attacks of a fairly dense frequency are aimed specifically at exhausting both our air defence forces and our physical and moral strength,” said Nataliya Gumenyuk, a spokeswoman for Ukraine's southern military command.
Ukrainian officials said most of the drones and missiles fired on Sunday and Monday had been shot down and President Volodymyr Zelenskyy praised US-supplied Patriot anti-missile defences.
“When Patriots in the hands of Ukrainians ensure a 100 per cent interception rate of any Russian missile, terror will be defeated,” Mr Zelenskyy said in his nightly video address on Monday.
Mr Vashchenko said on Monday he was not one of the few high-ranking Ukrainian officials who knew the exact date when the counter-offensive would be launched, but he said he strongly believed it would start in the “nearest future”.
“Nobody knows when the offensive will start, maybe three people know,” he said.