According to local media, the 47-year-old judge had sentenced a group of ultra-nationalists from the Russian fascist group known as the "white wolves" to up to 23 years in jail in February.
The group, comprising mostly teenagers, were found guilty of a string of brutal murders against migrants from Central Asian countries, many of whom had been bludgeoned to death.
Sova, a non-governmental organisation that tracks racist violence in Russia, said Chuvashov faced threats against him on several radical websites.
"They published his picture and extracts of audio tapes from court cases, profiling him as a danger to all Russians," Galina Kozhevnikova of the Sova Centre told AFP.
Investigators said the contract-style killing was likely to have been related to Chuvashov's work.
A police source told RIA Novosti that the suspect acted without frenzy, firing a "control" shot to ensure the judge was dead, and collected the spent gun shells before fleeing the crime scene.
Anna Usacheva, a spokeswoman for Moscow court, told the Echo of Moscow radio the judge was a "kindhearted, compassionate man and the highest of professionals".
His killing comes just over a year after the murder by right-wing radicals of a top lawyer who was also active in the fight against racism.
Two Russian nationalists were arrested and charged with the killing in January 2009 of human rights lawyer Stanislav Markelov and journalist Anastasia Baburova as they emerged from a news conference.
Russia's FSB security services said the two had been part of an extreme nationalist group, which was amassing weapons and had committed a racist murder.