Those killed in the exchange of fire were part of a cell that left dead seven pilgrims in Friday's attack in Minya province, a statement by the ministry said.
"The terrorist elements opened fire on the (security) forces who responded," the statement said.
The 19 were found "as part of a pursuit of terrorist elements involved in carrying out hostile operations in the country, including the last armed attack which targeted citizens returning from the Saint Samuel monastery", the ministry said.
The Islamic State group claimed Friday's attack in a message via its propaganda agency Amaq.
Raids were undertaken in the mountainous western desert of Minya province to track down the "fugitive terrorist elements", the interior ministry said.
Copts, a Christian minority that make up 10 percent of Egypt's 97 million people, have in recent years been repeatedly targeted by IS group. In May 2017, masked gunmen ordered Christians travelling to Saint Samuel to get off their buses and recant their faith.
The group refused and were shot one by one, leaving 28 people dead in the IS-claimed attack.
IS also killed more than 40 people in twin church bombings in April 2017, and an IS gunman last December killed nine people in an attack on a church in a south Cairo suburb.
A Coptic Christian woman mourns victims killed in an attack a day earlier, during an early morning ceremony at the Prince Tadros church in Egypt's southern Minya province.
Nineteen suspected jihadists linked to a deadly attack on Coptic Christians in central Egypt have been killed in a shootout with police, the interior ministry said Sunday.
