Shehab said "After contacts between the Islamic Jihad leadership and the brothers in Egypt it was agreed that a comprehensive ceasefire will begin immediately".
"We have seen and established a clear link between Gaza and Damascus", he said, adding that Israel's response will not be "limited geographically".
The Israeli military also shot and killed four Palestinians and wounded 232, 180 of them from live fire, during protests along the border with Gaza on Friday, Palestinian medical officials said.
It said its air force hit about 80 targets including a four-storey building used by Hamas as a headquarters, in response to more than 30 rockets launched into Israel.
Eight different alarm sirens were activated throughout the night in Israel's southern communities, including the city of Sderot near the Gaza border.
Confrontations between Gaza and Israel began escalating in late March after Palestinians launched a campaign of weekly Friday protests against Israel.
Egypt wants to restore calm in order to pursue the broader goal of Palestinian reconciliation between Hamas and the West Bank-based administration of President Mahmoud Abbas.
The Gaza protest campaign has included frequent attempts to cut through the 10-foot-high (3-meter) wire border fence to enter Israel. Health officials in Hamas-ruled Gaza reported that Israeli troops killed five Palestinians, and a sixth died while making a homemade bomb.
A Palestinian protester uses a slingshot to throw stones at Isreali forces during a demonstration at the Israel-Gaza border, east of Khan Yunis in the southern Gaza Strip on August 24, 2018.
Protesters are demanding the right of return to their homes and villages in historical Palestine from which they were driven in 1948 to make way for the new state of Israel.
Israel rarely acknowledges it has reached agreements with terror groups in Gaza.
Israel has struck scores of times inside Syria during the seven-year civil war there, at what it has said were Iranian targets or Teheran's transfers of weapons to Hezbollah fighters. COGAT, the Israeli defense body that handles Palestinian civilian issues, said one of the rockets hit an ambulance crossing used to transfer Gazans out of the strip in emergency humanitarian cases.
Israeli Prime Minister Benjamin Netanyahu made a rare visit to Oman on Friday.
The White House confirmed on Thursday that Mideast envoy Jason Greenblatt will arrive in Israel next week to address the Israel-Palestinian conflict.