Islamic Jihad said the raid was carried out by two of its al-Quds fighters and two more from Al-Aqsa Martyrs Brigades, an armed group loosely linked to Fatah.
Hours later, Israel targeted the Islamic Jihad offices.
The army said a separate air strike hit a weapons production facility. Residents said a dairy truck was hit but no one was hurt.
Fresh fighting meanwhile erupted between rival Palestinian factions in the Gaza Strip, killing a Hamas gunman and wounding dozens from both Hamas and the rival Fatah movement, hospital officials said.
Hamas and Fatah each accused the other of sparking the gun battle in the southern town of Rafah, and both sides deployed hundreds of men in the streets as the fighting wore on for several hours.
In the course of the battle, Hamas blew up a house after pummelling it with rocket-propelled grenades and machine-gun fire, accusing Fatah fighters of holing up inside, witnesses said.
The severity of the fighting made it hard for ambulances to reach the wounded, residents said.
The raid on the Israeli checkpoint was the first such infiltration since June 2006, when Gilad Shalit, an Israeli soldier, was captured.
Hiba Akila, Al Jazeera's correspondent in Gaza, said the Kissufim attack has been dubbed the "hot summer" operation.
It was called a retaliation for Israeli practices against the Palestinians, she said.
The attack came just hours after Israeli tanks and troops entered the southern Gaza Strip, near the town of Rafah.
Israeli army soldiers fought members of the armed wing of Hamas, the army and Hamas said.
"Hamas is trying to draw us [Israel] into their civil war"
Zafon, Beersheva, Israel
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Four Palestinians were arrested during the Israeli army raid.
An Israeli military spokesman confirmed on Saturday morning that an incursion by tanks and infantry units was under way east of Rafah.
"The military came under fire. They retaliated, hitting an armed man," the spokesman said, without elaborating.
Hamas confirmed the clash in a statement but made no mention of casualties.
The Israeli military detained dozens of Palestinians aged between 15 and 50 for questioning and identity checks, witnesses said.
In the West Bank, a 17-year-old Palestinian was reported to have been killed and another injured by Israeli fire near the West Bank city of Hebron.
Residents of the area said that Israeli soldiers opened fire at two young men who went hunting near the city.
The residents said that the victims were not affiliated with any Palestinian faction.
The Israeli army admitted it opened fire at the two young men, claiming that the presence of armed men in the area was a threat that should be dealt with.
Meanwhile, medics at a northern Gaza Strip hospital walked out of work on Saturday to protest against the kidnapping and "kneecapping" of a doctor by Palestinian armed groups.
Fayez al-Barrawi, a known Hamas supporter, was blindfolded, handcuffed and shot six times in the legs, including a kneecap, and then tossed on the street on Thursday.
Hamas has said Fatah fighters were behind the kidnapping.
Al-Barrawi worked at Beit Hanoun hospital in the northern Gaza Strip, close to the border with Israel.
It is the first time a doctor has been seized. However, fighters often fire on paramedics and ambulances during gun battles.