The situation in the Amhara - located in northwestern part of Ethiopia and is one of nine regional states in the country - is now under full control of the Federal Government, the statement added.
In the Amhara attack, the regional president Dr Ambachew Mekonne, a regional government officer adviser, Ezez Wassie were both killed.
Violence between the Amhara and Gumuz ethnic groups left dozens of people dead last month in Amhara and its neighbouring region, Benishangul Gumuz.
Ethiopia's army chief and the president of a key region have been shot dead in a wave of violence highlighting the political instability in the country as Prime Minister Abiy Ahmed tries to push through reforms. Within a few hours, all access to the Internet had been shut down and army and police checkpoints sprung up around the capital. Over a million people have been displaced by ethnic clashes, which analysts attribute to multiple causes, such as the weakening of the once all-powerful ruling EPRDF and different groups trying to take advantage of opportunities presented by the political transition.
In his TV address, the premier said that the botched coup was led by a very high-ranking regional military official and others within the country's military. It is not known if he has been arrested, though many others involved in the plot have been apprehended.
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Internet services went down as the coup attempt was taking place, according to monitoring group NetBlocks, which said national IP connectivity had dropped to 2 percent of normal levels, making it hard to assess the current situation.
His government has released political prisoners, removed bans on political parties and prosecuted officials accused of gross human rights abuses.
Ethnic bloodshed - long held in check by the state's iron grip - has flared up in many areas, including Amhara, where the regional government is led by Ambachew Mekonnen.
The US state department has warned its staff in Addis Ababa to stay inside, saying it is aware of reports of gunfire in the city on Saturday.
Residents in Bahir Dar reported hearing heavy gunfire. "Hence, it is not evident whether the internet shutdown is being implemented by those who organized the uprising, or by government forces who are seeking to suppress it".
Ethiopia, a nation of 100 million people, is due to hold a national parliamentary election next year.