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.
