Tension between the government and the Tigray People's Liberation Front (TPLF), which controls Tigray, boiled over into military clashes this month.
The rocket attack damaged "airport areas", the government said. The TPLF has not confirmed responsibility, but has said "any airport used to attack Tigray" is a "legitimate target".
It comes after reports of a massacre.
Human rights group Amnesty International said it had confirmed that "scores, and likely hundreds, of people were stabbed or hacked to death" in the town of Mai-Kadra (May Cadera) on 9 November.
Prime Minister Abiy Ahmed has accused forces loyal to Tigray's leaders of carrying out the mass killings, while the TPLF has denied involvement.
Mr Abiy ordered a military operation against the TPLF earlier this month after he accused them of attacking a military camp hosting federal troops - claims the TPLF deny. There have since been a number of clashes and air strikes in the region.
The conflict has forced thousands of civilians to cross the border into Sudan, which says it will shelter them in a refugee camp.
Getting independently verified information about the situation in Tigray is hard because phone lines and internet services are down.
What do we know about the rocket attack?
The Ethiopian government's emergency task-force said rockets were fired towards the cities of Bahir Dar and Gondar, in Amhara state, late on Friday.
"The TPLF junta is utilising the last of the weaponry within its arsenals," it wrote, adding that an investigation had been launched.
An official told Reuters news agency that one rocket hit the airport in Gondar and partially damaged it, while a second one fired simultaneously landed just outside of the airport in Bahir Dar.
Details on casualties were not immediately clear.
Both airports are used by military and civilian aircraft.
Forces from the Amhara have been fighting alongside their federal counterparts against Tigray.
The rocket attack has raised fears that fighting in Ethiopia's northernmost state could spread to other parts of the country.
Were there mass killings in Tigray?
Amnesty said evidence showed that "scores" of people were killed and wounded in knife and machete attacks in Mai-Kadra.
It said it had seen and "digitally verified gruesome photographs and videos of bodies strewn across the town or being carried away on stretchers".
Amnesty said the victims appeared to be labourers not involved in the conflict. It is not clear where they came from.
Ethiopia's human rights commission said it would send a team to investigate the reports.
Mr Abiy has accused forces loyal to Tigray's leaders of the massacre, saying they went on the rampage after federal troops had "liberated" the western part of Tigray.
Some witnesses also said the attacks were carried out by forces loyal to the TPLF after they had been defeated by federal troops in an area called Lugdi.
Tigray leader Debretsion Gebremichael told AFP news agency that the accusations were "baseless".
The UN human rights chief warned the killings could amount to war crimes.
Why are the Ethiopian government and TPLF fighting?
Tension has been mounting for some time as relations between the TPLF and the federal government have deteriorated.
The TPLF dominated Ethiopia's military and political life for decades before Mr Abiy took office in 2018 and pushed through major reforms.
Last year, Mr Abiy dissolved the ruling coalition, made up of several ethnically-based regional parties, and merged them into a single, national party, the Prosperity Party, which the TPLF refused to join.
Tigray's administration sees Mr Abiy's reforms as an attempt to build a unitary system of government destroying the current federal arrangement.
It also resents what it calls the prime minister's "unprincipled" friendship with Eritrean President Isaias Afwerki.
Mr Abiy won the Nobel Peace Prize in 2019 for his efforts to bring peace with long-standing foe Eritrea.
For his part, the prime minister believes the TPLF officials are undermining his authority.
Mr Abiy ordered the military operation against the TPLF after he said its fighters had crossed "the last red line".
He accused them of attacking a military camp hosting federal troops on 4 November, calling the action "treasonous". The TPLF has denied attacking the camp.
What is life like in Tigray?
By Hana Zeratsyon, BBC Tigrinya
Communication is difficult at the moment because internet and mobile phone services have been cut.
There are already reports of a shortage of flour and fuel - and, worst of all, water, which was already rationed.
In Mekelle, which has a population of between 400,000 and 500,000, homes used to get piped water once a week, but the supply has stopped.
Families used to buy water from vendors, but with phones disconnected they can no longer call to put in orders.
On Thursday it was reported that a power-generating dam had been damaged in an air strike, cutting electricity supply in the region.
I am anxious about the safety of my family, especially 11-year-old brother who is suffering from cerebral palsy and epilepsy.
I am extremely worried about whether he will get his medication.
As he cannot talk, I used to see him during video calls but that is now not possible.
Read more about life in Tigray