The State Duma lower house of parliament passed the controversial measure in a 436 to 0 vote with one abstention, clearing the way for its expected passage by the upper house and signature by President Vladimir Putin.
The bill bans "propaganda of non-traditional sexual relations" to minors by Russians and foreigners as well as media organisations.
The measures makes it an offence to say that gay relationships are equal to heterosexual ones.
"Traditional sexual relations are relations between a man and a woman," one of the bill's authors, Yelena Mizulina of the left-leaning, Kremlin-friendly A Just Russia party, told lawmakers.
"These relations need special protection from the government."
Mizulina said the bill's authors had earlier referred to "homosexual relations" but they introduced the softer wording for the second and third readings expected later Tuesday.
If individuals use media or Internet for such "propaganda" they can be fined up to 100,000 rubles ($3,000), while organisations can be fined up to one million rubles and can be closed down for up to 90 days.
The bill also targets foreigners, apparently because international gay rights activists such as Britain's Peter Tatchell have regularly travelled to Russia to support gay pride events.
It says that foreigners who use media or the Internet for propaganda can be fined up to 100,000 rubles and can also be held in police cells for up to 15 days and be deported.
Rights activists and Western governments have repeatedly criticised the bill.