About 20 armoured vehicles, accompanied by bulldozers and two helicopter gunships, early on Tuesday moved 1.5km inside the Hamas-governed territory near the Kissufim crossing point with Israel.
They searched a school and other buildings in El Karara and Wadi al-Salka.
Exchanges of fire took place between Israeli soldiers and Palestinians and a number of explosions were reported.
Other sources said Israeli aircraft overnight raided the Maghazi refugee camp in the centre of the Gaza Strip.
The raid targeted members of the DFLP, but no one was injured.
A military spokesman in Tel Aviv was unable to confirm those reports.
Earlier, Palestinian medical workers identified the DFLP activist assassinated late on Monday as Ibrahim Abu Olba, 42.
Witnesses said he was killed when an Israeli air strike targeted a vehicle in northern Gaza.
Dr Moaiya Hassanain of the Palestinian health ministry said Abu Olba died of his wounds in a Gaza hospital.
Three bystanders were also hurt, he said.
Israel normally supplies 60 per cent of the electricity for Gaza’s 1.5 million inhabitants
Gaza needs around 240MW of electricity, but normally receives only about 200MW, with 8 per cent from Egypt
Israel is the only source of industrial fuel for Gaza’s power station
Israel first stopped supplying industrial fuel supplies to Gaza on January 19
The EU pays Israel around $10m per month for Gaza’s industrial fuel
The Israeli military confirmed that it carried out the strike targeting Abu Olba, who it said had been involved in a string of attacks against Israel and was planning more.
Separately, the Israeli defence ministry said it would renew fuel shipments for Gaza’s power plant on Wednesday, but not for cars and other vehicles.
Cooking gas will also be allowed into Gaza, the ministry said.
Israel cut off fuel shipments to Gaza after Palestinian fighters killed two Israelis at the only fuel transfer depot to Gaza last week.
“We receive fuel usually on a daily basis, but since last week we have received very little fuel,” Rafiq Maliha, director of the Gaza power station company, told Al Jazeera on Monday.
“If the fuel supply is not resumed as usual we have to shut down the power plant in a couple of days.
“The Gaza power plant serves at least half a million people living in the Gaza Strip, it also serves all the main infrastructure including the water supply, hospitals, treatment plant, everything, so this is almost stopping life.”