The move comes after evidence of state-sponsored doping in sports saw Russia's track-and-field Olympics team and entire Paralympics squad banned from the Rio Games this summer.
The law -- which still needs to be approved by the senate and President Vladimir Putin to come into force -- includes a one-year jail term for officials who incite a minor to dope or use violence or threats to force an athlete to take illegal drugs.
Russia is struggling to mend the damage to its battered reputation after reports from the World Anti-Doping Agency (WADA) lifted the lid on systemic cheating in the country.
Moscow denies any state-run doping programme but has pledged to toughen up its tainted anti-doping measures.
Its track-and-field athletes remain banned from international competition over the doping accusations.
The legislation does not outline any punishments for convicted drug cheats but Dmitry Svishchev, a member of the parliament's sports and physical culture committee, told AFP he would submit a separate bill that could see doped athletes face criminal charges.
"I can now say that all our obligations regarding our anti-doping programme have been fulfilled," he said.
Alexander Zhukov, the president of the Russian Olympic Committee who also holds a seat in parliament, told local news agencies that the law demonstrates the government's "absolute intolerance" for doping.
"This is of course an answer to our foreign critics who accused our country of allegedly having a state programme that supports doping," Zhukov said.
The second part of a bombshell WADA report covering doping across Russian sport is expected to be released next month.