“We’ve been able to enter Mekelle city without innocent civilians being targets,” Mr. Abiy was quoted as saying by Ethiopian Broadcasting Corporation.
The city of half a million braced for an all-out offensive by government forces against its dissident leaders.
Mr. Abiy, who won last year’s Nobel Peace Prize, announced on November 4 he had ordered military operations against Tigray’s ruling party, the Tigray People’s Liberation Front.
Thousands of people have been killed in more than three weeks of fierce fighting and tens of thousands have streamed over the border into Sudan. Displacement within Tigray is thought to be widespread.
Mr. Abiy announced on Thursday he had ordered a “final” offensive and Ethiopia’s military said it had encircled Mekelle.
A communications power cut in Tigray has made it difficult to verify claims about how the fighting is going.
A spokesman for a crisis committee formed in response to hostilities in Tigray did not respond to a request for comment about reported shelling in Mekelle, which has already been hit by airstrikes.
But Addis Ababa on Saturday said the “aerial engagement has been precise and targets only TPLF’s military depot, weaponry, and arsenals”, avoiding “civilian facilities”.
The Tigray government accused Mr. Abiy of teaming up with Isaias Afwerki, the president of neighbouring Eritrea, for the assault on Mekelle.
“The Tigray regional state would like it to be known to friends and enemies alike that it will give a proportional response to the massacres and property damages being done by those fascists,” it said.
The TPLF dominated Ethiopian politics for nearly three decades before Mr. Abiy came to power in 2018, and fought a border war with Eritrea between 1998 and 2000 that killed tens of thousands.
Mr. Abiy won his Nobel Prize in large part for initiating a rapprochement with Mr. Isaias in 2018.
Ethiopia has denied enlisting Eritrean military support against Tigray but has acknowledged using Eritrean territory.
Residents of the border city of Humera in western Tigray said that shells fired from Eritrea hit both residential and commercial structures during fighting earlier this month.
At least one rocket fired from Tigray was aimed at Eritrea’s capital, Asmara, on Friday night, regional diplomats said. There were no reports of casualties or damage.
The TPLF claimed responsibility for rockets fired towards Asmara two weeks ago, but there was no immediate claim of responsibility for the latest attack.
Global concern has heightened in recent days with world leaders and human rights groups sounding warnings of possible rules of war contraventions during the operations.
The UN has spent weeks lobbying for full access to Tigray, so far unsuccessfully.
Mr. Abiy’s office said this week it would open a “humanitarian access route”. Hundreds of UN and international NGO workers are currently in Mekelle, but they are grappling with shortages of food, cash, and other essentials.
The International Committee of the Red Cross said it had received 1,300 requests from people in Ethiopia and abroad frantically trying to contact their relatives.
“We know this is just the tip of the iceberg,” it said.
Pope Francis on Saturday tweeted for “everyone to pray for #Ethiopia where armed clashes have intensified and are causing a serious humanitarian situation”.
Mr. Abiy’s government has said the military campaign in Tigray was triggered by attacks by pro-TPLF forces on federal army camps in Tigray that began on the night of November 3.
He has repeatedly snubbed international calls for a halt to fighting and negotiations with TPLF leaders, saying they must be disarmed and apprehended.
On Friday he met three former presidents of African countries – Joaquim Chissano of Mozambique, Ellen Johnson Sirleaf of Liberia, and Kgalema Motlanthe of South Africa – sent by the African Union as mediators.
The AU said Mr. Abiy had told envoys that military operations in Tigray “would not last long”.
The military expects to take control of Mekelle “within a few days”, according to a report on Saturday from the state-affiliated broadcaster FBC.