The truck blew up inside an Israeli base and fighters inside the jeep began attacking soldiers with light arms and hand grenades, the statement said.
In the course of the clashes, an Israeli watchtower was blown up, it added.
But the Israeli army denied that Palestinian fighters had infiltrated into Israel or attacked a post.
Israeli aircraft, though, destroyed two cars in the area around the time the clashes are said to have occurred.
Islamic Jihad later said that six Palestinian fighters had been killed on Thursday by Israeli helicopter fire during an attack on an Israeli military post near the border with the central Gaza Strip.