Most of the perpetrators of the attack in Bahir Dar, capital of Amhara, have been apprehended and others are being hunted down, said the spokesman.
The attack in Bahir Dar and the assassination of the army chief are linked, he said.
The US Embassy reported gunfire in the capital Saturday and urged people to be careful.
In Bahir Dar residents told The Associated Press they could still hear gunshots late Saturday. Ethiopia's internet appeared to be shut down on Sunday.
The attempted coup in Amhara is the latest challenged to Abiy, who was elected last year as a reform-minded young leader.
Abiy has captured the imagination of many with his political and economic reforms, including the surprise acceptance of a peace agreement with bitter rival Eritrea, the opening of major state-owned sectors to private investment and the release of thousands of prisoners including opposition figures once sentenced to death.
Along the way he has faced some challenges. In June 2018, only months in office, an attempt to hurl a grenade at Abiy caused a deadly explosion at a massive rally in support of the sweeping changes in Ethiopia. Nine police officials were arrested over that incident, state media reported.
In October rebellious Ethiopian soldiers protested over pay, causing a security incident in the capital. Ethiopia's army is one of the largest in Africa.
The country is a key regional security ally for the United States and others in the restive Horn of Africa region.
- AP