Eritrean forces spotted the drive through the Ethiopian town of Edigaret, 25 km from the border, according to Reuters citing witnesses and ACLED.
This is the first confirmed presence of Eritrean troops on Ethiopian soil since the current conflict began on September 23.
ACLED says they crossed at Zalambasa and an Ethiopian military drone attacked their vehicles at Gwada.
Eritrea denies this and calls the reports fabricated to justify the Ethiopian aggression.
Ethiopia and Eritrea severed diplomatic ties last week.
The US called on both sides to exercise restraint, and warned that a conflict would jeopardize US cooperation with the two countries.
There is growing fear of a wider war in the Horn of Africa that will include Ethiopia, Eritrea, Sudan and Egypt.