Rwanda has accused soldiers from Democratic Republic of Congo of "provocation" after a group crossed the border, leading to clashes along the volatile area.
One of the Congolese soldiers was shot dead, while a Rwandan was wounded in the exchange of fire, a Rwandan military spokesman said. A Congolese spokesman said the clash came after the Congolese soldiers crossed the border to buy beer.
Rwanda denies backing DR Congo rebels. Some 500,000 people have fled their homes in eastern DR Congo since April, when the M23 group took up arms.
The UN has accused Rwanda of backing the group - Rwanda twice invaded its much larger neighbour during the 1990s and was accused of looting eastern DR Congo's rich mineral resources.
Rwandan military spokesman Brig Gen Joseph Nzabamwita said soldiers from DR Congo's armed forces (FARDC) had crossed the border on a reconnaissance mission, reports Reuters news agency.
"This is a provocative act of the FARDC. They want to draw Rwanda into their internal problems but Rwanda has always said we will not escalate the problem," he said.
Congolese army spokesman Col Olivier Hamuli said a Rwandan soldier had also been killed but this was denied by Brig Gen Nzabamwita, reports the AFP news agency. The clash is said to have occurred near Kibumba, 13km (8 miles) north of the Congolese city of Goma.
Rwanda is widely seen as having backed armed groups in eastern DR Congo as a way of tackling Hutu fighters who fled there after the 1994 genocide. Both Rwanda and the M23 rebels, now known as the Congolese Revolutionary Army, are led by ethnic Tutsis.