Protesters remained on the streets late into the night following the vote, with student Josh Hakim saying he was "really, really sad about what's happening to this country."
"You see what is happening on the streets, everyone is so angry," he told AFP at a rally near the parliament in Jerusalem.
Some 58 people were arrested at demonstrations, the police said, among them protesters in Tel Aviv, which has become the focal point of one of the country's largest-ever protest movements.
Police said one person was arrested for allegedly harming demonstrators, with protest organisers saying he drove a car into people blocking a highway.
Officers used water cannon to disperse protesters on a major road through Tel Aviv, where the crowd waved Israeli flags.
Opposition slams 'puppet' PMÂ
Netanyahu failed to appease opponents with a televised address late Monday, in which he pledged to hold talks during the upcoming parliamentary recess. "Reach a comprehensive agreement on everything and we will add more time should it be needed," he said.
The embattled premier showed signs of fatigue in the chamber, as he sat between his defence and justice ministers just a day after unscheduled surgery to fit a pacemaker.
Netanyahu defended the new law, which limits the powers of the Supreme Court in striking down government decisions, as a "necessary democratic step."
Deep divisions within his own coalition and mass protests prompted the premier to temporarily halt the legislative process in March, but within weeks politicians were blaming each other for the breakdown in negotiations.
On Monday, the opposition walked out of the chamber to boycott the vote, which passed with 64 votes in the 120-seat chamber.
Opposition chief Yair Lapid slammed Netanyahu's "unprecedented performance of weakness."
"There is no prime minister in Israel. Netanyahu has become a puppet of messianic extremists," he said, a reference to the premier's far-right and ultra-Orthodox Jewish allies.
The political instability has raised alarm among Israel's allies abroad.
The White House described the vote as "unfortunate". A German foreign ministry source said: "We look with great concern at the deepening tensions in Israeli society."