The incident is the first apparent breach of the January 18 ceasefires.
The Israeli military said the blast came from a bomb planted by Palestinian fighters targeting a patrol near the Kissufim crossing into Gaza.
The military said that an officer and two other soldiers were also wounded in the attack.
Palestinian medical workers said that after the incident, Israeli forces opened fire, killing a Palestinian farmer.
Dr Moaiya Hassanain of Gaza's health ministry confirmed that a 27-year-old Gaza farmer was killed by Israeli gunfire along the border, but several miles away, said.
Two other Palestinians were wounded.
Israel subsequently closed its crossings into Gaza to humanitarian aid traffic after briefly opening them in the morning.
Raed Fattouh, a Gaza border official, said Israeli officials informed him the closure was due to the attack.
Describing the attack on the Israeli patrol, Tamer Mishal, Al Jazeera's correspondent, reporting from Gaza, said an anti-armour shell was fired from inside the territory at an Israeli tank near the area of Khan Younis.
"Witnesses told Al Jazeera that the grenade directly hit an Israeli jeep," he said.
"Palestinian residents reported the sound of gunfire and Israeli helicopters in the area."
The residents of Kissufim said the Israeli army patrol and Palestinian fighters exchanged fire shortly after the blast.
Ayman Mohyeldin, Al Jazeera's correspondent in Gaza, said no Palestinian group has yet claimed responsibility for the attack.
"The likely scenario is that an improvised explosion device of some sort was under that patrol as it was driving and detonated ... but some reports suggest it was a mortar round that was used against the patrol," he said.
"We are not sure if it was remotely detonated or if a pressure sensor caused it to trigger, but we do know that as a result of that explosion one Israeli solder was killed and three are in a critical state of injury.
"Immediately after a large blast of gunfire was heard in the area, there were several farmers out on their land, and we understand that one Palestinian farmer was killed."
Israeli troops withdrew from Gaza last week after a three-week offensive whose stated aim was to stop Palestinian rocket fire into southern Israel.
Egyptian mediators have been talking separately to Israel and Hamas to negotiate a more permanent ceasefire.
Hamas wants the border crossings into Gaza reopened, including the Rafah crossing into Egypt, to end the Israeli blockade in the territory.
Israel wants to stop the rocket fire and prevent Hamas fighters from using smuggling tunnels under the border with Egypt to rearm themselves with weapons.
Barack Obama, the US president, has sent George Mitchell, his newly appointed Middle East envoy, to the region to discuss the ceasefire efforts.
Obama instructed Mitchell, who played a prominent role in the Northern Ireland peace process, to "engage vigorously" to achieve real progress between Israel and the Palestinians.
In an interview to Dubai's Al Arabiya television, Obama praised Saudi King Abdullah for putting forward in 2002 a plan for peace in the Middle East, known as the Arab peace iniative.