Russia says it is pulling back some of its troops from near Ukraine after a build-up raised fears of an invasion.
The defence ministry said large-scale drills continued but that some units were returning to their bases.
Ukraine warned to wait to see proof of the pull-out, saying “when we see the withdrawal, then we’ll believe the de-escalation”.
More than 100,000 Russian troops have massed at Ukraine’s border. Russia has always denied it is planning an attack.
The build-up has brought increasingly grave warnings, with the US saying an invasion could come at anytime.
Russia has been seeking guarantees that Ukraine will not be allowed to join Nato, something the security bloc has rejected.
In its statement, Russia’s defence ministry said it was withdrawing some of the troops conducting exercises in military districts bordering Ukraine.
“A number of combat training exercises, including drills, have been conducted as planned,” Defence Ministry spokesman Igor Konashenkov said.
Some exercises are continuing, such as a large joint Russia-Belarus drill, due to end on 20 February.
A British government source said it was waiting to see the scale of the withdrawal, saying it would have to make a difference to the ability to invade to be meaningful.
But the announcement was enough for both Ukraine and Russia to claim victory in the stand-off.
Ukraine’s Foreign Minister Dymytro Kuleba said “we have managed together with our partners to deter Russia from any further escalation”.
A Russian foreign ministry spokeswoman, Maria Zakharova, said the day “will go into history as the day western war propaganda failed. They have been disgraced and destroyed without a single shot being fired.”
Diplomatic efforts aimed at warding off a potential crisis continue on Tuesday, with Germany’s Chancellor Olaf Scholz due to meet Russia’s President Vladimir Putin.
He is likely to repeat warning that Russia faces tough sanctions if it invades Ukraine.
Mr Scholz has faced criticism for his response to the tensions. He is not expected to scrap the Nord Stream 2 gas pipeline, which will bring Russian gas to Western Europe, despite US President Joe Biden saying the scheme would be halted if Russia invades.
