State-run Rossiya 24 TV channel said the assailant was armed with a rifle and had begun shooting on the street before moving into a nearby store.
The car was later found by police, who identified the suspect as a man of around 30 years of age with a criminal record.
Special forces officers were deployed to the address the suspect is believed to be holed up at, a law enforcement source told RIA Novosti.
Police and the city administration said a 14-year-old girl was among the dead, five of whom died at the scene of the attack. The sixth victim died shortly after in hospital.
Media reports suggested a second suspect was involved in the attack, but this was not confirmed by police.
The attack comes some six months after a Moscow lawyer shot dead six people in the Russian capital in what was believed to have been his violent response to the end of a romance.