He wrote on Twitter that the "Big Brother law" was an "unworkable, unjustifiable violation of rights that should never be signed".
Among the new rules are tough punishments for failing to report crime, or inciting terrorism online.
It must still be signed into law by Russian president Vladimir Putin.
Mr Snowden, a former contractor for the CIA, fled to Russia in 2013 after leaking details of extensive internet and phone surveillance by the US National Security Agency.
Commenting on the law, he wrote: "Mass surveillance doesn't work. This bill will take money and liberty from every Russian without improving safety."
The new rules - known as "Yarovayan's legislation" after politician Irina Yarovayan - state:
- telecoms companies must keep copies of customers' phone calls and text messages for six months
- phone and text records (but not the messages themselves) must be kept for three years
- online services (such as social networks) must keep message records for one year
- online services that encrypt data must help security services decrypt any message sent by users, or face a fine
- failing to report knowledge of a crime will become a criminal offence - punishable by up to a year in prison
- inciting or expressing approval of terrorism online will be regarded as publishing such comments in mass media - punishable by up to seven years in prison
- children aged over 14 can be held criminally liable for 10 new charges such as taking part in terrorism
However, rules that would let law enforcement revoke Russians' citizenship, or revoke people's right to leave Russia, were removed from the legislation.
The legislation has been criticised by opposition politicians. Yuri Sinelshchikov warned that storing telecoms data could lead to abuse by officials, while Dmitry Gudkov said the laws would be a heavy burden on businesses.
"Instead of competing and entering new markets and improving connection quality, our telecommunications companies will have to deal with this stupidity," he said.
Mr Snowden said asking companies to store six months of communication data was "not just dangerous, it's impractical".
Russia's State Duma approved the final draft of the legislation on Friday. The country's Federation Council must approve it before President Putin signs it into law.