"I hope that it (anti-smoking law) will contribute to reducing this factor (smoking)," Medvedev said at the first session of the government's health protection committee.
Russia has some of the highest rates of smoking in the world.
President Vladimir Putin signed the ban into the law Feb 25. It envisages a gradual ban on smoking in public, including in government buildings, healthcare and educational facilities, cultural sites, sports stadiums and on public transport.
The law comes into force June 1, 2013, though some of its provisions, including a total ban on smoking in restaurants, trains and hotels, will only come into effect a year later.
Medvedev cited expert data according to which the ban will "not only save health but also up to 200,000 lives" a year, which is equal to "the population of a large Russian city."
The legislation also imposes a total ban on tobacco advertising. Tobacco companies will be barred from holding promotional prize draws and sponsoring public events and competitions.
The bill also prohibits display of tobacco products in shops. Sales in retail kiosks will be banned from June 2014, and minimum prices will be set for cigarettes, which currently cost from just 40 rubles (about $1) a pack.
Russia signed the Framework Convention on Tobacco Control with the World Health Organisation in 2008. This obliges Russia to enact tougher restrictions on tobacco by 2015.