Two civilians, a beduin and his child, were killed in an explosion near the Nahal Oz kibbutz in southern Israel on Tuesday.
The army said the blast was either caused by a fresh rocket attack or a dormant rocket that exploded.
Abu Bara, a spokesman for Islamic Jihad, told AFP: "We carried out this attack in order to disrupt the Israeli elections.
"We strike [Israel] to show the world that this state is illegal and we will carry on."
Abu Bara said Jihad - the faction behind the last seven anti-Israeli bombings - fired three makeshift rockets towards Israel from an area near Karni, the sole transit point for goods entering and leaving Gaza.
The projectiles were fired towards the Nahal Oz area. He said another two rockets were fired from the northern Gaza Strip into Israel.
Palestinian militants frequently fire makeshift rockets into southern Israel from the Gaza Strip, but many fall short or do not initially explode on impact.
Hardline Israeli parties have criticised frontrunner Ehud Olmert's plan to withdraw unilaterally from parts of the West Bank, saying it would reward and invite more Palestinian violence.
Olmert's opponents have pointed to the rocket attacks that have continued out of Gaza since Israel withdrew from the territory last summer.