"An Israeli soldier... was killed earlier this morning," an army statement said.
"A terrorist who was identified infiltrating Israel from the southern Gaza Strip opened fire at Israeli soldiers, who responded with fire. During the exchange of fire, the terrorist was killed," it said.
The Palestinian killed was identified as member Ahmed Nasser, 20.
But while the movement's military wing Al Quds Brigades "congratulated the heroic operation," it said the movement had "no connection to the action."
The father of Ahmed Nasser, bearing the same name, told AFP his son was a member of the group.
"I'm very proud of Ahmed's heroic operation," said the farmer. "Ahmed had hoped to carry out such an operation, or kidnap Israeli soldiers. He spoke about it several times."
"I'm sad to have lost a son, but happy he's a martyr," he added.
The Israeli soldier was named as Nitnel Moshiashvili, 21, of the southern city Ashkelon.
Following the incident, Israeli aircraft and tanks fired warning shots into Gaza. Palestinian sources said the shelling, east of Khan Yunis, resulted in the burning of a three-hectare (7.4-acre) wheat field.
Late on Friday morning, Palestinians fired two rockets from Gaza into Israel, a military spokeswoman told AFP. There were no reports of casualties or damage.
And on Friday afternoon, three Palestinians were wounded in an Israeli air raid.
Medical sources in Gaza said the attack targeted Palestinians on a motorcycle east of Khan Yunis, seriously wounding three members of the Popular Resistance Committees.
One of the wounded died of his wounds later Friday, the same sources said.
Witnesses said that following the exchange of fire, Israeli soldiers backed by tanks entered Palestinian territory east of Khan Yunis and captured one person, and then retreated. The Israeli army could not confirm the report.
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