The bills -- the second of which would ban loudspeakers in urban areas between 11:00 pm and 7:00 am -- will eventually have to be reconciled, with three more readings required before they can become law.
They were approved after a heated discussion that turned into shouting matches between ruling coalition members and Arab lawmakers, some of whom tore copies of the legislation and were ejected from the chamber.
The bills passed 55-48 and 55-47 in the Knesset, or parliament.
While the bills in theory would apply to any religious place of worship, Muslims say it is clearly meant to silence the traditional call to prayer at mosques.
The measure has become commonly known as the "muezzin law" after the Muslim official charged with calling the faithful to prayer, often through powerful speakers mounted on minarets.
The notion of Israeli legislation silencing mosques has sparked outrage around the Arab and wider Muslim world.
Supporters of the move say it is needed to prevent daily disturbance to the lives of hundreds of thousands of non-Muslim Israelis.
Last month, government ministers endorsed the softer version of the bill prohibiting loudspeakers overnight, which limits its scope to the first of the five daily Muslim calls to prayer just before dawn.
That version would apply to mosques in annexed east Jerusalem as well as Israel, but not to the highly sensitive Al-Aqsa mosque compound, Islam's third holiest site, according to an Israeli official.
'A racist act'
An earlier draft limiting volumes throughout the day had been rejected because it might have silenced the siren sounded in Jewish areas at sunset on Friday to mark the start of the Sabbath.
However, the stricter measure was revived by members of the hardline Yisrael Beitenu party, part of Prime Minister Benjamin Netanyahu's coalition, leading to Wednesday's two votes.
It was not immediately clear if that version would apply to Al-Aqsa, located in mainly Palestinian east Jerusalem.
One of the sponsors of the less rigid bill, Motti Yogev of the far-right Jewish Home, said the proposal was "a social law that aims to enable people to sleep".
"Loudspeakers have not been here forever, and in recent decades there are alarm clocks for whoever wants to wake up for the mosque," he said.
Ahmad Tibi of the predominantly Arab Joint List alliance of lawmakers called the measure "a racist act".
"This is an important Muslim religious ceremony, and (the Knesset) has never intervened in a Jewish religious event," he said.
Opposition has not only come from Arabs and Muslims.
Israeli President Reuven Rivlin has spoken out against the move, saying existing noise pollution regulations provide a solution.
Government watchdog groups have called the measure an unnecessary provocation that threatens freedom of religion.
At Wednesday's debate, Environmental Protection Minister Zeev Elkin said the new law was necessary since the existing rules set a low fine that causes police to disregard noise violations.
The new proposed law sets a fine of 10,000 shekels ($2,714, 2,573 euros) to transgressors.
In Jordan, the official custodian of Muslim holy sites in annexed east Jerusalem, Information Minister Mohamed Momani condemned the bills as "discriminatory".
They were contrary to "Israeli commitments under the peace accord" that the Jewish state signed with Jordan in 1994, he said, quoted by the official news agency Petra.