Ethiopian forces killed a Sudanese army officer and wounded others when they opened fire on a border patrol by its north-western neighbour on Monday.
It was the latest deadly clash between the two sides in an increasingly tense dispute.
On Tuesday, the Sudanese military said its border force returned fire when attacked, killing all the Ethiopian assailants. It said one of the patrol’s two vehicles was destroyed.
At the core of the border tension between Sudan and Ethiopia is a move by Sudanese troops in December to wrest back control of border enclaves inside Sudan.
The disputed areas had long been settled by farmers from Ethiopia’s powerful Amhara group who are attracted by the fertility of the soil in the area.
The farmers enjoyed the protection of federal forces and allied militias.
Sudan has since vowed to retake other border enclaves held by the Amhara, while Ethiopia said it will not negotiate with Sudan unless it pulls out from the enclaves it retook.
Khartoum has rejected that condition and vowed to regain control of other areas still held by the Amhara militias and farmers.
The clash on Monday prompted one of Sudan’s top generals, Yasser Al Atta, to travel to the border region on Tuesday to meet military units stationed there.
Lt Gen Al Atta is also a member of the Sovereignty Council, a 14-member body that functions as a collective presidency and is led by Sudan’s top general Abdel Fattah Al Burhan.
Earlier this week, the Sudanese Prime Minister Abdalla Hamdok told a senior EU envoy in Khartoum his country had no intention of going to war with Ethiopia over the border dispute.