The country’s one-day increases in new cases have fallen to their lowest levels since April.
Over the past 24 hours, 93 people have died, bringing the total toll to 9,166 — a rate considerably lower than in many other countries hit hard by the pandemic.
A total of 4,343 people recovered over the last 24 hours, bringing the overall number of recoveries to 403,430.
Moscow lifted restrictions including compulsory travel passes on June 8, a move welcomed by residents who rushed out to enjoy the streets and parks.
Many other regions are lifting lockdown restrictions as Russia prepares to hold a July 1 national vote despite the pandemic.
President Vladimir Putin is seeking public backing for constitutional amendments that would allow him to stay on at the Kremlin beyond his current term.
Critics have cast doubt on Russia’s low official mortality rate and accused authorities of under-reporting to play down the scale of the crisis.
Russia attributes its lower virus death figures to mass testing which has identified many cases with mild or no coronavirus symptoms.
The Health Ministry is now adjusting how it reports numbers to include all deaths believed to be related to the virus even if the direct cause of death was another condition or the patient tested negative.
Former Finance Minister Alexei Kudrin, a liberal politician known for his outspoken comments, said Russia’s health system requires more funds and modernization in many regions.
“Medicine is significantly underfunded even according to the current norms and also it needs a serious overhaul,” Kudrin, who heads the Audit Chamber that examines government spending, said in an interview with the state-run TASS state news agency.
“We will carry out a check and show these figures,” he added.
Nevertheless, the economist said Russia “is not doing a bad job with the pandemic,” crediting the role of the military, which has built emergency facilities.