But the Russian leader will be fully aware of the magnitude of the threat of Western-supplied long-range missiles.
The Institute for the Study of War, a think tank, has published a map of 225 Russian military installations within range of ATACMS.
Former US envoy to Ukraine, Kurt Volker, said Biden’s decision would enable Ukraine “to go after airfields, the ammunition depots and the fuels supplies, logistics that Russia has, which right now are in a sanctuary zone in Russia”.
Biden’s decision will cause Russia to be more cautious, Volker told the BBC.
Dismissing Putin’s threats, he said the Russian leader “should have anticipated that there would be efforts by Ukraine to fire back”.
Ukraine has had ATACMS as well as UK and French Storm Shadow missiles of similar range for some time, though the numbers are not known. But it has not been allowed to use them inside Russia.
France and the UK are expected to follow the US lead and issue the same authorisation to Ukraine. So far, they have not commented.
White House officials are emphasising to US media that Biden’s change of heart is in response to Russia’s deployment of North Korean troops – a signal to Pyongyang not to send any more.
It comes after Ukraine has endured a barrage of Russian attacks in recent days.
One strike on Odesa on Monday killed 10 people, including seven policemen, and injured 47 others.