She said that troops exchanged fire with Hamas fighters at several points, and identified hitting them.
In one battle, Israeli troops were "targeted by Palestinians and there were several casualties," she said, adding that there had been several incidents and that Palestinians had used anti-tank missiles and mortars as well as assault rifles.
Hamas officials said that the Israeli soldiers were supported by helicopter gunships during the fighting near the Nahal Oz border terminal.
Two Israeli civilians, two Palestinian civilians and two fighters were killed in attack on the border crossing last week, the only terminal for supplying fuel to the Gaza Strip and its 1.5 million people.
Fuel deliveries, which were halted after the April 9 attack, were resumed on Wednesday, Israeli and Palestinian officials said.
Israeli officials had said earlier on Wednesday that the terminal would not be reopened.
The move to resume fuel deliveries came a day after the UN called for Israel to re-open supplies so Gazans could undertake daily life.
Mahmoud Khuzandar, deputy director of the Gaza fuel station owners' association, said Israel had pumped 90 tons of liquid gas for cooking and 180 thousand litres of fuel for the power station into Gaza.
However, an Israeli defence official said that they had only sent fuel for a power plant.
Local residents said that as they withdrew, Israeli armoured vehicles hit and damaged a mosque.
Witnesses said orange and olive trees were also uprooted. The Israeli army did not immediately comment on the damage to the mosque and trees.
Rockets and mortars were fired into southern Israel during the fighting, according to the Israeli military.
Late on Tuesday, an Israeli air raid killed member of the Palestinian Islamic Jihad movement as he rode a motorbike near Jabaliya in the north of the territory, medics said.
Several bystanders were also wounded in the attack.
Meanwhile, in the West Bank at least 23 Palestinians were detained by Israeli forces overnight.