Later in the fighting, a Palestinian sniper fired from the border area into Israel, killing a volunteer from Ecuador who was working in a field at the communal farm of Ein Hashlosha, according to Israeli rescue services.
Israel's incursion comes a week after George Bush, the US president, said he wanted a peace deal between the Israelis and the Palestinians by the end of his presidential term, at the end of 2008.
An Israeli army spokeswoman said troops opened fire on a group of armed Palestinians who were allegedly approaching them.
Israel said they were engaging in an operation "against terror threats".
Taher Nunu, a Hamas spokesman, responded to the attacks by saying: "What is clear is that there is an Israeli plot to destroy the whole region.
"To all the Arab countries, to the custodian of the two Holy mosques [King Abdullah of Saudi Arabia], Arab leaders and heads of states, we would like to tell them that they have to act immediately in order to put an end to this aggression.
"President Bush gave the green light to such aggression and such terrorism," Nunu said.
"All of us are ready to be martyred. We are living martyrs ... Our objective is to liberate the Palestinian people," he said.
Most of those killed in the Israeli incursion were activists of Hamas' Izz al-Din al-Qassam Brigades.
David Chater, Al Jazeera's correspondent in Jerusalem said the "major" Israeli incursion began late on Monday night.
He said: "We understand that Apache helicopter gunships, armoured bulldozers, tanks and ground troops are all involved in this incursion."
"There are also reports of another incursion in the industrial zone near the Erez crossing."
"Ehud Olmert [the Israeli prime minister], has reportedly ordered a series of "sharp and short" incursions."
Hussam al-Zahar, son of al-Zahar, a former Hamas-appointed foreign minister, was killed in the Israeli operation.
In 2004, al-Zahar's eldest son was killed in a failed Israeli attempt to assassinate him.The deaths follow Monday's missile attack by Israel near the Gaza home of Ismail Haniya, the Hamas leader, which killed three Palestinians.
Tuesday's deaths bring to 6,067 the number of people killed since the start of the second Palestinian uprising in September 2000, the vast majority of them Palestinians, according to a news agency tally.
'Suspension of talks'
|Al-Zahar's eldest son was also killed |
by the Israelis in 2004 [AFP]
Meanwhile, al-Zahar has called for the suspension of Israeli-Palestinian talks initiated by President Bush during his visit to Israel and Palestine last week.
He said: "We believe that this Zionist escalation is the result of Bush’s visit and a natural result of the cover supplied by Bush to the Israeli occupation authorities to go forward in their escalation and aggression against our Palestinian people, particularly in Gaza Strip."
"We will retaliate in the way they understand."