"This is a step towards our attempts to break the siege in the face of this criminal Zionist enemy."
Israel said it killed fighters driving the other two vehicles before they had a chance to detonate.
Clashes then broke out between Israeli troops and Palestinian fighters.
Statements on Izz al-Din al-Qassam Brigades' website said the operation was aimed at capturing Israeli soldiers.
Saturday's attack was the second such incident at a cargo crossing from Israel into the Gaza Strip in the last 10 days.
Jacky Rowland, Al Jazeera's correspondent in Gaza, said: "The Israeli army described the attack as complex, and clearly well planned in advance.
"This is clearly a new tactic that we're starting to see by the military wing of Hamas and the wing of Hamas has itself said that it plans to carry out more of these kinds of attacks in the coming days."
Hamas identified the dead as Mahmud Abu Smara, Ahmed Abu Salenan and Ghasen Arhaem, all in their 20s.
|An Israeli strike in Rafah left one Palestinian dead and three others injured [AFP]|
Within hours of the attack on the crossing, Israel launched an air raid in southern Gaza, targeting a car in Rafah.
One Palestinian - identified as a mechanic for the Hamas police force - was killed in the strike, and at least three other people injured, Palestinian security officials and medics said.
A member of Izz al-Din al-Qassam Brigades had been killed overnight and another injured in an Israeli air raid east of Gaza City, Palestinian medical sources and the Israeli military said on Saturday.
The slain fighter was identified as Imad Abu Amar, aged 22.
Separately, the Israeli military said rockets fired from Gaza temporarily knocked out electricity in the border town of Sderot.
The rockets were fired into Israel late on Friday by members of the Islamic Jihad group.
Palestinian fighters launch crude rockets at Israeli border towns almost daily.
The renewed violence in Gaza came shortly after Jimmy Carter, the former US president, met Khaled Meshaal, the exiled Hamas leader, in the Syrian capital Damascus, and the Israeli government announced plans to build 100 new homes in two illegal settlements in the occupied West Bank.