The clashes, injuring four police and six protesters, have become an almost nightly occurrence since Ramadan started nearly two weeks ago.
Palestinian militants in the Gaza Strip fired around three dozen rockets into Israel overnight on Saturday, while the Israeli military struck back at targets operated by the ruling Hamas group.
Meanwhile hundreds of Palestinians clashed with Israeli police in east Jerusalem, injuring at least four police and six protesters.
The clashes have become a nightly occurrence throughout the Muslim holy month of Ramadan and show no signs of stopping.
Tor Wennesland, the UN envoy to the region, called for calm and condemned the violence.
"The provocative acts across Jerusalem must cease. The indiscriminate launching of rockets towards Israeli population centres violates international law and must stop immediately,'' he said.
"I reiterate my call upon all sides to exercise maximum restraint and avoid further escalation, particularly during the Holy month of Ramadan and this politically charged time for all.''
Jerusalem, home to holy sites sacred to Jews, Christians and Muslims, has long been a flashpoint in the Israeli-Palestinian conflict.
In 2014, similar tensions erupted into a 50-day war between Israel and Gaza's ruling Hamas militant group.
The Israeli military said a total of 36 rockets were fired into Israel throughout the night. Six rockets were intercepted, the military said, while others landed in open areas.
There were no reports of injuries or serious damage, but the incoming rocket fire set off air-raid sirens throughout southern Israel.
In response, the army said fighter jets and helicopters struck a number of Hamas targets in Gaza, including an underground facility and rocket launchers. Hamas did not claim responsibility for the rocket fire, but Israel considers the group responsible for all fire emanating from the territory.
The military imposed limits on outdoor gatherings in southern Israel early Saturday but lifted the restrictions several hours later and allowed people to resume their normal routines.
Israel and Hamas, an Islamic group sworn to Israel's destruction, are bitter enemies that have fought three wars and numerous skirmishes since Hamas seized control of Gaza in 2007.
Although neither side appears to have an interest in escalating tensions, Hamas sees itself as the defender of Jerusalem and may feel obligated to act, or at least tacitly encourage rocket attacks by other groups, ahead of upcoming Palestinian parliamentary elections. Hamas' armed wing has warned Israel "not to test'' its patience.
At dawn, hundreds of people in Gaza challenged nightly curfews imposed by Hamas to curb the coronavirus outbreak and took to the streets in an act of solidarity with fellow Palestinians in Jerusalem, burning tires.
The Palestinians want east Jerusalem to be the capital of their future state. Its fate has been one of the most divisive issues in the peace process, which ground to a halt more than a decade ago.
Palestinians have clashed with Israeli police on a nightly basis since the start of the Muslim holy month of Ramadan nearly two weeks ago.