Russia's overall case tally, the world's second highest after the United States, rose by 8,894 to 326,448, while Friday's tally of 150 new deaths pushed the toll to 3,249.
Moscow's Mayor Sergei Sobyanin has said he expects the toll to prove much higher in May than in April. It is still much lower than many European countries with fewer cases, something that has sparked debate, but which Russia defends.
Speaking at a televised government meeting by video link, Putin said there had been a gradual decrease in the number of new infections reported daily in recent days and that the situation "as a whole is stabilising".
"At the same time it is very important this decline happens against the backdrop of a constant and significant rise in the number of tests," he said.
Russia's national virus data shows a steady decline in the number of reported cases, but the Kremlin's critics and Russian media outlets have questioned the veracity of some of the figures.
A poll of 509 doctors on an app called the "Doctor's Handbook" cited by the Meduza news website on Thursday found that more than a third of those surveyed had been ordered to attribute pneumonia deaths caused by COVID-19 to other causes.
The health ministry did not immediately respond to a Reuters request for comment on the report.
The survey found almost 40% of medical professionals said they had inadequate supplies of personal protective equipment, while 23.4% said their hospitals were not ready to admit patients with suspected coronavirus infections.
Moscow began mass testing residents for virus antibodies this week. Of 40,000 asymptomatic people tested, 14% showed traces of infection in their blood, according to a report by Open Media, a news outlet funded by Kremlin critic Mikhail Khodorkovsky.