"Soldiers patrolling along the Israel-Egypt border identified a number of suspects who had infiltrated into Israel," the military said in a statement.
"The suspects fired at the soldiers, who returned fire and identified a hit," the statement added, saying all but one member of the group escaped back into Egypt.
"Upon the arrival of an IDF (Israeli Defence Force) medical force to the location of the incident, they determined the death of the injured suspect."
The incident is the latest in a string of apparent attempts by militants to infiltrate Israel via its border with Egypt, according to the Israeli military.
On 21 February, troops interrupted an apparent smuggling attempt during which a man hurled a bag containing explosives at soldiers.
The bag failed to detonate and the suspect was able to escape, the army said.
And on 23 February, in the same border area where the latest shootout occurred, Israeli troops discovered a bag containing a large explosive device after unsuccessfully attempting to detain an infiltrator.
Border security incidents have increased in recent months as Israel has warned that lawlessness in post-revolution Egypt is allowing militants to use Sinai to stage attacks against the Jewish state.
It accuses militants based in Gaza of using the long border between Egypt and Israel to infiltrate and carry out attacks, including one last August in which gunmen launched coordinated ambushes on vehicles in Sinai, killing eight Israelis.
Israel is building a giant steel security barrier along the frontier, which is due to be completed by the end of this year.