The proposal -- denounced by critics as likely to encourage a witch-hunt against leftist groups that campaign for the defence of Palestinian rights -- was passed in a 50-43 vote following a tense debate.
Two more readings of the bill by the parliament, or Knesset, are required for it to become law.
The text has renewed tension between one of the most rightwing governments in Israeli history and the United States and the European Union.
Justice Minister Ayelet Shaked, who proposed the law, argues it will boost transparency as the government seeks to fight foreign interference and attempts to delegitimise the state of Israel. She has insisted it does not target any specific NGO.
Prime Minister Benjamin Netanyahu has also expressed support for the law.
The text does not specifically refer to leftist organisations, but they are expected to be most affected as right-wing NGOs, such as those supporting Israel's occupation of the West Bank, tend to rely on private donations, particularly from within the United States.
Several leftwing Israeli NGOs receive large percentages of their funding from abroad, including from European governments.
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The bill has sparked international criticism, with the US and EU ambassadors to Israel both expressing concern over its implications.
It stipulates that NGOs receiving more than half of their funding from foreign governments will have to declare it in all their official reports.
At the request of Netanyahu, however, a proposed requirement for NGO members to wear a badge indicating their organisation is funded by a foreign country was dropped.
"I do not understand how a requirement for transparency is anti-democratic; the opposite is true," Netanyahu said last month. "In a democratic regime, we need to know who is financing such NGOs, from the right, the left, up or down."
Israeli leftwing NGOs have expressed concern at increasingly personal attacks they have been subjected to in recent months, including regular harassment and even death threats.
Settlement watchdog Peace Now has called the bill "a hate crime against democracy" while the United States has warned it could have a "chilling effect".
In a statement on Tuesday, Peace Now said "the passing of the NGO bill is a violent and discriminatory act of public shaming against those criticising the government.
"Despite Netanyahu's statements, the bill resembles the situation in Russia and not that in the United States or in any other democratic country," it said.