Police also say 200 people were involved in the clashes, which are thought to have broken out in a row over jobs.
The clashes broke out at the Khovanskoye cemetery in southwest Moscow.
More than 90 people were arrested after officers intervened to break up the fighting crowds.
Many of those involved were armed with bats, sticks and metal bars.
Reports said police officers fired shots into the air in an attempt to quell the violence.
Police and Interior Ministry officers made 50 arrests.
According to the police, fighting broke out during disputes between ethnic groups from ex-Soviet Central Asia and Russia’s volatile North Caucasus, who were competing for some of the cemetery’s lucrative services business.
Police spokeswoman Sofya Khotina said the victims died after being run over by others trying to escape the scene in their cars.
The cemetery covers a 500-acre site on Moscow’s southwestern rim.
