Followers of the Orthodox Church - the most widely practised faith in the country - were observing part of the Epiphany celebrations on Thursday when clashes broke out with youths from the neighbouring Oromia region.
The number of casualities could increase as witnesses say some people are unaccounted for.
The government has vowed to ensure those behind the violence are prosecuted.
The Epiphany festival, known in Ethiopia as Timket, has continued with a heavy security presence. The event has Unesco's World Heritage status.