The shootout occurred in the desert outside Ras al-Sidr, a Red Sea town 40km south of Suez, when police confronted a group of suspects that they had been pursuing since Monday evening, the ministry said in a statement on Tuesday.
Security officials said the group of beduins - one of whom was killed - were suspected of having links to last year's deadly bombings of Sinai resorts.
The ministry statement identified the dead man as Muhammad Abd al-Rahman Badawi.
They did not confirm earlier reports that said two people were killed, a policeman as well as a beduin. One police officer was wounded, the ministry said.
Sources told Aljazeera that a group of beduins in Ras al-Sidr had tipped off security forces about three people suspected of carrying explosives, adding that the surviving two members of the group are being hunted down.
Officials at Suez hospital named the policeman as Ibrahim Subhi Ibrahim, 32, and said his wounds were not life-threatening.
"They opened fire on a police vehicle, resulting in an exchange of fire"
Egyptian interior ministry statement
Authorities said Badawi was a suspect in the bombing of one of two tourist camps in Nuwaiba, south of Taba on the Sinai's east coast. He was armed with a rifle and pistol, the ministry added.
Security forces, acting on a tip-off, had deployed to the mountainous central Sinai region where the men were believed to have been and sealed it off in preparation for a raid, security sources said.
But the suspects pre-empted the attack and opened fire on them before fleeing further into the mountains.
"They opened fire on a police vehicle, resulting in an exchange of fire," the interior ministry said.
It added that security forces were pursuing the "fugitives".
Human-rights groups said security forces arrested thousands of suspects in the wake of the attacks on 7 October on the Taba Hilton and the two camps in Nuwaiba that killed 34 people and wounded 105.
But authorities have announced the arrests of only a handful of suspects.
The explosion at the four-star hotel was believed to have been caused by a powerful car bomb detonated near the entrance to the building, destroying an entire wing of the 10-storey block.
On Friday, police clashed with about 500 demonstrators who were protesting against the state's three-month detention of people suspected of involvement in the bombings. The clash, in which three police officers and about 10 protesters were injured, took place in the north Sinai town of al-Arish.