Three other officers were wounded in Tuesday's attack in South Sinai province at a checkpoint near the ancient site, the ministry said.
The shooting, which the ministry said on Tuesday was carried out by "a number of gunmen", was claimed by the Islamic State group.
"Security forces in South Sinai, in cooperation with bedouin elements tracked the perpetrators' escape route and closed roads that would be used to escape," the scene of the shooting, the ministry said in a statement.
It said the suspect began firing after he was found by security forces, "which required the forces to quickly deal with him, injuring him leading to his death."
Security staff found an automatic weapon and ammunition, adding that "investigations are ongoing to identify the terrorist."
St Catherine's monastery, a Christian site 500km southeast of Cairo in the south of the Sinai attracts thousands of visitors a year.
Tuesday's attack comes nine days after Palm Sunday services at Coptic Christian churches in the cities of Tanta and Alexandria were struck by twin bomb attacks claimed by ISIS, killing 45 people.
ISIS has threatened to carry out more attacks on Copts, which makes up about 10% of Egypt's population of more than 90 million people.
Egypt has battled a jihadist insurgency in North Sinai province that has killed hundreds of policemen and troops since the 2013 ouster of Islamist president Mohamed Morsi.
In October 2015, ISIS claimed the bombing of a Russian airliner carrying holidaymakers from the popular South Sinai resort of Sharm el-Sheikh, killing all 224 people on board.