In the State Duma lower house, 441 deputies voted for the measure with one against, Russian news agencies reported.
The Kremlin headed by President Vladimir Putin has identified smoking as a critical public health issue that causes up to 400,000 deaths annually in a country where over 40 percent of the adult population are smokers.
After the State Duma's approval, the bill just needs approval from the upper chamber and a signature by President Putin and could become law by June 1.
The bill introduces strict no-smoking rules in most public areas, including schools, medical and sports facilities, beaches, passenger trains and ships, and establishes a non-smoking perimeter around entrances to train and metro stations, usually areas thick with smoke.
Smoking is also to be prohibited in restaurants, cafes and hotels.
The law also sets up new conditions for selling tobacco, banning sales in small kiosks, and establishing minimum prices per pack. Currently cigarettes cost as little as $1.30 and are sold everywhere, often to teenagers.
Tobacco companies will also have to adhere to new rules banning sponsorship of sports, culture, or education events, and hiding cigarette packs from view in stores, according to the bill.
Russian Prime Minister Dmitry Medvedev vowed last October to halve the number of smokers in the country, lamenting that it did not stand up to Big Tobacco corporate giants in the early 1990s, becoming the world's second largest market after China.
"Unfortunately, the government did not consider the risks of foreign tobacco investment into the Russian economy in the 1990s," he said in a video blog, decrying the growing number of female and underage smokers.
Neither Putin nor Medvedev are smokers. However Medvedev is surrounded by several chain-smoking ministers in the cabinet such as Foreign Minister Sergei Lavrov.
Lower life expectancy caused by smoking costs the budget an estimated 1.2 trillion rubles ($40 billion), or 6.3 percent of the country's GDP, Oleg Salagai, a spokesman for the health ministry, told Interfax news agency.
Tobacco giants have strongly criticised some of the bill's propositions, like the ban on laying out cigarette packs openly in store displays or selling them in kiosks.
British American Tobacco said the bill has "several reasonable suggestions," but decried some clauses that it deemed "not entirely realistic and therefore excessive," predicting bankruptcy for many kiosk owners, whose main earnings come from selling cigarettes.
In a poll by the independent Levada Centre last year, 22 percent of respondents said they smoke at least a pack per day. Thirty-eight percent of those in the 18 to 24 age group said they smoke on a daily basis.
Fifty-six percent said they "fully support" the smoking ban, while 15 percent said they don't support it.
Putin has not spoken publicly against smoking. But he has made a healthy population one of his biggest priorities and called on Russians to adopt healthy habits in his annual state of the nation address last year.