JERUSALEM — A Grad rocket fired from the Gaza Strip struck in southern Israel early Tuesday, threatening to further escalate tensions that have been mounting since Saturday, when a 30-year-old Palestinian prisoner died in an Israeli jail.

The rocket, which came down on a road outside the city of Ashkelon and caused no injuries, was the first from Gaza to hit Israel in the three months since a cease-fire agreement ended eight days of cross-border violence. Israel has violated the cease-fire several times by firing on fishermen and farmers approaching newly relaxed security perimeters, but the agreement has otherwise held.

A subgroup of Al Aksa Martyrs Brigade, the military wing of the Palestinians’ Fatah faction, said it fired the rocket in response to what it called the “assassination” of Arafat Jaradat, the prisoner who died on Saturday. Palestinian leaders have placed blame for Mr. Jaradat’s death on what they described as “severe torture” during interrogation, though Israeli officials say the bruising and broken ribs cited as evidence of torture could have been caused by resuscitation efforts.

Mushir al-Masri, a lawmaker from the militant Hamas faction that rules the Gaza Strip, said in an interview that Israel was “fully responsible for the consequences of the wave of the Palestinian public fury.”