The Ethiopian forces had been forced to retreat from the Umm Barakit area, the military said, without giving further details.
The head of Sudan's military, General Abdel Fattah Al Burhan, told reporters the incident took place on Saturday. He said it showed how the military was protecting the country after a coup attempt in Khartoum last week.
Col Getnet Adane, Ethiopia's military spokesperson, did not immediately respond to a request for comment.
Tensions along the border have escalated since the outbreak of a conflict in Ethiopia's northern Tigray region last year, that sent tens of thousands of refugees into eastern Sudan. A dispute has boiled over a border area of fertile farmland, known as Al Fashaqa.
Tens of thousands of refugees have streamed across the border into Sudan since war broke out 10 months ago in Tigray.