The Sudanese army also accused Ethiopia of displaying the bodies of the executed people to the public and vowed that there would be "an appropriate response".
"This treacherous act will not pass without a response," it said.
There was no immediate response from Ethiopian authorities to a request for comment.
Tensions between Sudan and Ethiopia have run high in recent years because of a spillover of the conflict in Ethiopia's northern Tigray region and Ethiopia's construction of a giant hydropower dam on the Blue Nile.
Tens of thousands of refugees have fled into eastern Sudan and there have been military skirmishes in an area of contested farmland along the border between Sudan and Ethiopia.