Hamas, which denied involvement in firing some of what Israel counted as eight rockets shot just as an earlier truce expired on Wednesday, and accused the Jewish state of violating the new truce by launching air strikes.
There were no reported casualties in any of the incidents that marred an Egyptian-brokered agreement announced in Cairo to extend a ceasefire begun on Monday by another five days, or until Aug. 19.
Israel had no comment on the deal the Palestinians announced in Cairo. Egyptian mediators had won the deal to extend a ceasefire when the sides were clearly headed toward failure to bridge key differences in time for a midnight deadline.
Prime Minister Benjamin Netanyahu ordered Israeli forces to fire rocks in response to what he called a breach of the ceasefire by Hamas.
Hamas official Izzat Reshiq denied the Palestinians had breached the truce, and denounced Israel's air strikes as "a violation of the calm."
The Israeli military said its air strikes were "targeting terror sites across the Gaza Strip," and these attacks were followed by two more rocket attacks at Israel from Gaza.
In announcing the truce extension on Wednesday, Azzam Al-Ahmed, the head Palestinian negotiator in Egypt, a member of Palestinian President Mahmoud Abbas's mainstream Fatah faction said on Wednesday evening in Cairo that "it was agreed to extend ceasefire by five days."
The extension was intended to give the sides more time to reach a more lasting deal after they had failed to bridge differences over an Egyptian proposal for a permanent truce that addressed a key Palestinian demand to lift the Israeli and Egyptian blockades of the Gaza Strip.
It was unclear how Israel's and Egypt's security concerns about Islamist Hamas, the dominant force in Gaza, were addressed by Egypt's new proposal, or how it could be reconciled with Israel's demand for Gaza's demilitarisation.
Hamas leader Ismail Haniyeh told Al-Aqsa Hamas television on Wednesday that the group would insist on "lifting the Gaza blockade" and reducing movement restrictions on the territory's 1.8 million residents, as a prerequisite to a "permanent calm".
Steps to ease blockade
Egyptian and Palestinian sources said Israel had tentatively agreed to allow some supplies into the Gaza Strip and relax curbs on the cross-border movement of people and goods, subject to certain conditions.
A Palestinian demand for a Gaza seaport and reconstruction of an airport destroyed in previous conflicts with Israel has also been a stumbling block, with the Jewish state citing security reasons for opposing their operation.
The sides have agreed to delay discussion of any agreement on the ports for a month, a Palestinian official said.
As part of the Egyptian blueprint, Israel was expected to expand fishing limits it imposes on Gaza fishermen to 6 miles (10 km) from the usual 3-mile offshore zone.
"It will increase gradually to no less than 12 miles in coordination between the Palestinian Authority and Israel," the official said, referring to a likely expanded role in Gaza affairs for the government of Western-backed Abbas, based in areas of the West Bank not occupied by Israel.
In addition, the official said, the Egyptian plan calls for reducing the size of a "no-go" area for Palestinians on the Gaza side of the border from 300 meters (328 yards) to 100 meters so that local farmers can recover plots lost during security crackdowns.
Israel and Hamas have not met face-to-face in Cairo: Israel regards Hamas, which advocates its destruction, as a terrorist group. But the official said once they inform Egypt of their agreement, a ceasefire accord could be signed the same day.
The Gaza hostilities have killed 1,945 Palestinians in the Gaza Strip and 67 on the Israeli side. Most of the Palestinian dead have been civilians, hospital officials in the small, densely populated enclave say.
Israel launched its military campaign on July 8 to quell cross-border rocket fire from Gaza
The heavy losses among civilians and the destruction of thousands of homes in Gaza - where the United Nations said 425,000 of a population of 1.8 million have been displaced by the war - have stoked international alarm.
Israel pulled ground forces out of Gaza last week after it said the army had completed its main mission of destroying more than 30 tunnels dug by militants for cross-border ambushes. It now wants guarantees Hamas will not use any reconstruction supplies sent into the enclave to rebuild the tunnels.