Estonians view the monument as a reminder of 50 years of Soviet occupation.
Russia, which has had troubled ties with Estonia since the 1991 collapse of the Soviet Union, had said moving the monument was an insult to those who fought against fascism.
Protest broken up
Police charged protesters, fired tear gas and rubber bullets and used water cannon on Friday night to break up protesters, many in their early teens.
|"We demand the Estonian authorities make available to us all information about the incident, investigate it promptly"|
Russian Foreign Ministry
"The situation calmed down at 2am [0100 GMT] on Saturday morning after police dispersed the crowds, and has been peaceful from that time," said Taavi Kullerkupp police spokesman.
About 50 premises, mostly shops, were vandalised on Saturday, compared with about 100 the day before.
Baltic news agency BNS said some disturbances involving youths had broken out on Friday in the town of Johvi, in the northeast, where many Russian speakers live.
Vladimir Putin, the Russian president, expressed his concern about removing the monument and the police crackdown on riots in a telephone conversation with Angela Merkel, German chancellor.
"In response, Merkel spoke in favour of finding a prompt solution to the situation and of both sides exercising restraint," a Kremlin press release said.
Russia's foreign ministry said in a statement the victim was a Russian citizen living in Estonia.
The statement said: "We demand the Estonian authorities make available to us all information about the incident, investigate it promptly and bring to justice those responsible for this crime.
"As a result of the excessive use of force by the Estonian authorities against the demonstrators ... tens of civilians suffered."