The al-Nejashi Mosque was reportedly shelled. Its dome, the minaret and tombs of Islamic figures were damaged.
The government said a nearby church damaged during the conflict would also be repaired.
Locals believe al-Nejashi was built by the first Muslims to migrate to Africa during the time of Prophet Muhammad.
They had fled persecution in Mecca and were given refuge in what was then the Kingdom of Aksum.
Local Muslims believe that 15 disciples of Prophet Muhammad are buried in the damaged tombs.
They also say the mosque is the oldest in Africa, though others believe that title belongs to one in Egypt.
The mosque is near Wukro town, some 800km (500 miles) from Ethiopia's capital, Addis Ababa.
An Orthodox Christian church named Saint Emmanuel was also damaged, but further details are unavailable.
A Turkish aid agency launched a project in 2015 to renovate the mosque, saying it wanted to "preserve the heritage" of the monument and wanted it to become a major destination for "religious tourism".