Responding to the latest violence, Israel's defence minister issued a veiled threat to Hamas.
"We have been used to seeing breakaway (Palestinian) groups doing the firing, and Hamas trying to calm things down. Possibly it is loosening its grip, for all sorts of reasons," Ehud Barak told Israeli television on Friday.
"Should that indeed prove to be the case, then there will also be ramifications for Hamas," he said, but added: "We have no interest in returning the region to what was in the past."
"An Israeli army force raided 500 metres into Palestinian territory, and was confronted by our gunmen," Abu Obeida, a Hamas spokesman, said.
"This was our work, but was carried out for defence."
Al Jazeera's Barnaby Phillips, reporting from Gaza, said: "The al-Qassam Brigades gave a press conference and said that at 2:30pm local time an Israeli incursion began into Gaza. Israeli soldiers on foot, tanks and helicopters crossed into Gaza and the al-Qassam Brigades responded with sniper fire.
"Witnesses in the area are saying that what took place is now over; it is quiet and Israeli tanks have withdrawn.
"Palestinian medical sources have told us three injured civilians have been taken to a hospital nearby," he said.
Witnesses said Friday's exchange of fire began when an explosion, possibly caused by an anti-armour rocket fired from the nearby Palestinian town of Khan Younis, hit an Israeli army patrol on the central Gazan border.
Backed by tanks, the troops fired back at their assailants and entered Gazan territory, the witnesses said. Such pursuits are common practise for the Israelis, who try to maintain a buffer zone within the border fence off-limits to Palestinians.
Our correspondent said Hamas officials in Gaza acknowledge the increase in tension that is "in some ways related, of course, to the situation in Jerusalem".
"Hamas say that they are at pains overall not to incite the Israeli army at this point in time and that they don't want trouble," he said.
The witnesses also said that, during the fighting, Israeli soldiers took away a wounded comrade and helicopters came to the scene, apparently for medical evacuations.