Palestinian medical sources confirmed the death of 55-year-old Hasan Abu Khatla early on Sunday morning in the centre of Rafah.
He had been shot in the chest and died instantaneously.
An eight-year-old boy was also injured during the dawn raid. Two other men were arrested, but no names have yet been released.
Israeli military sources confirmed the shooting and told Aljazeera.net that troops had tried to arrest the man at his home. But they were forced to pursue him when he fled with two other people, one of whom was armed.
However, the official could not explain why Khatla was shot in the chest if he were being pursued by Israeli forces as he ran away.
His house was later demolished after being searched by the army, the Israeli source confirmed.
According to the UN relief agency UNRWA, Israel carried out an average of 74 house demolitions each month in the Palestinian territories during the first half this year.
The troops, who had entered Rafah in armoured vehicles, then pulled out of the town.
Last month, Rafah was the scene of a deadly Israeli incursion lasting more than a week that sparked sharp criticism from UN Secretary General Kofi Annan and was responsible for the deaths of dozens of men, women and children.
The latest death brings to 2696 the number of Palestinians killed since the second Intifada broke out in September 2000.