Russian law adopted in November 2012 requires that all NGOs engaged in political activity and receiving foreign funding register as "foreign agents".
The third reading is seen as a formality, after which the amendments will have to be passed by the Senate and then be signed into law by President Vladimir Putin.
The move by the parliament's lower house is a rapid response after the Russian state-funded TV channel RT registered with the US Justice Department as a foreign agent following pressure from the US government.
During Wednesday's debates, State Duma speaker Vyacheslav Volodin described the new legislation as a "symmetrical answer" to the USA and a signal that "our media can't be treated like that".
The broadly phrased bill will leave it to the Russian government to determine which media outlets would be designated as foreign agents, said Leonid Levin, the head of the Duma committee for information.
After acquiring this status, these media outlets will be subject to the restrictions and responsibilities, which are now envisaged for non-governmental organizations designated as foreign agents.
Outlets such as Voice of America and Radio Free Europe/Radio Liberty, which receive funding from the US Congress, would be forced to register as foreign agents.
The label would apply if the outlet is either registered overseas, receives foreign funding or gets paid by a Russian company that is itself financed from overseas, the State Duma announced on Tuesday.
RT capitulated to a second US deadline on Monday to comply with an anti-Nazi propaganda law that requires the broadcaster to disclose its financing and label itself as a foreign agent. At the same time, the websites that those organisations use have not been blocked in Russian Federation, even though those resources "are used to prepare and disseminate materials that discredit Russia's domestic and foreign policies, to form negative public opinion and destabilise the situation in the country", the law-makers said.