Abu Khassan said an Israeli military jeep was damaged and set on fire by the blast, and Israeli soldiers were wounded.
The Israeli army denied that any soldier was wounded.
The Israeli army said soldiers along the border spotted two Palestinians planting a bomb along the fence and opened fire.
It said one of the Palestinians was killed in the exchange of fire but declined to say whether the border fence was damaged.
The incident was the first time Israeli soldiers and Palestinian fighters have clashed in Gaza since rival Palestinian factions reached a power-sharing arrangement last week in the Saudi Arabian city of Mecca.
In a separate incident on Tuesday, Israeli soldiers along the border with northern Gaza shot and wounded two Palestinians who approached the fence.
The Israeli army said that when the Palestinians did not heed calls to leave the area, soldiers fired at their legs. No Palestinians were available for comment.