The army officer was a border guard and the two men were from the Central Security force, the official said. The three were killed as the Israeli military chased militants along the border of Egypt's Red Sea resort of Taba in South Sinai and the Israeli city of Eilat.
"An Israeli plane had been chasing militant infiltrators along the border between Taba and Eliat and one Egyptian Central Security officer and two Central Security men were caught in the line of fire," the army official said.
Three other security men were injured by gunfire and were being moved to a hospital in the city of el-Arish in North Sinai, a security source in South Sinai said.
The Sinai forms a huge desert buffer zone between Egypt and Israel, which sealed an historic peace treaty in 1979 after fighting two wars in less than a decade.
Earlier Thursday gunmen killed seven people in southern Israel in attacks along Egypt's porous border, prompting Israel to chase infiltrators along the border and launch an air strike in the Gaza Strip that killed six Palestinians.
Israel said the attackers infiltrated from the Hamas-run Gaza Strip via Egypt's Sinai desert, despite stepped-up efforts by Egyptian security forces in recent days to rein in Palestinian and Islamist radicals.
The terms of the Camp David accords signed by Egypt and Israel in 1978 help explain why it is difficult to police Egypt's borders and maintain control in Sinai, where well-armed Bedouin occasionally clash with security forces.
The accords, agreed by former Egyptian President Anwar Sadat and Israeli Prime Minister Menachem Begin, demilitarized central Sinai and allowed Egypt to deploy only a small number of lightly armed border guards there and on the 266-km (165-mile) frontier.
After Israel pulled out of Gaza in 2005, Egypt proposed raising the number to 3,500 to help it secure its border with the Gaza Strip. Israel refused, citing security concerns.
Egypt's Central Security force is used to quell protests and guard the Sinai border along with border patrols.