Some people died in a panicked stampede after police employed tear gas, rubber bullets and baton charges, they said.
Thousands had gathered for a religious festival in Bishoftu, 40km (25 miles) from the capital Addis Ababa.
Some reports said police responded after anti-government protesters threw stones and bottles.
Others said demonstrators were entirely peaceful.
Prime Minister Hailemariam Desalegn blamed "evil forces" for the deaths in a national address on state TV.
He said rioters had caused "pre-planned mayhem" that led people to fall to their deaths in ravines. He vowed to bring to justice those responsible.
The prime minister denied reports that the security forces had opened fire, and praised their "great efforts" to protect the public.
There has been a series of deadly clashes in Ethiopia in recent months.
People in the Oromia and Amhara regions have complained about political and economic marginalisation.
The US has expressed concern about what it termed the excessive use of force against protesters.
Crowds at Sunday's Oromo festival, which AP news agency said had attracted two million people, chanted "We need freedom" and "We need justice", witnesses said.
Some participants crossed their wrists above their heads, a gesture that has become a symbol of Oromo protests.
The unrest was sparked last November by a plan to expand the capital into Oromia. This led to fears that farmers from the Oromo ethnic group, the largest in Ethiopia, would be displaced.
The plan was later dropped but protests continued, highlighting issues such as marginalisation and human rights.