The official says guards first fired a warning shot as the man tried to cross the barbed wire fence early Saturday. When he failed to stop, he was shot dead.
Police have stepped up efforts to control the border with the Israel since May, after what had been a six-month lull in known fatalities and after what security sources say is a rise in human trafficking through Egypt.
At least 16 migrants have been killed since May, with the last case reported in September.
The border is a main transit route for usually unarmed African migrants and refugees seeking work or asylum in Israel. Egyptian police complain that the smugglers who ferry the migrants to the border region sometimes fire on security forces.
Egypt, which for years tolerated tens of thousands of African migrants on its territory, is under Israeli pressure to halt the illegal trafficking.
Egyptian authorities also say they are concerned the flow of migrants at its Sinai border could pose a security threat in an area where Islamist militants sometimes find refuge.
Analysts and aid workers say the flow of migrants from the Horn of Africa through Egypt to Israel has increased in recent months as it has become more difficult to travel on other northward routes, such as via Libya to Europe.
Eritreans make up the single largest group of people attempting to cross into Israel from Egypt, although Ethiopians and Sudanese also make the trek.