The admission in a letter to the UN Security Council - and posted online by Eritrea's Ministry of Information - comes a day after UN aid chief Mark Lowcock said the world body had not seen any proof that Eritrean soldiers have withdrawn.
Eritrea has acknowledged its troops are participating in the war in Ethiopia's northern Tigray region but has vowed to pull them out amid mounting international pressure.
The first explicit admission of Eritrea's role in the fighting came in a letter posted online Friday night by the country's information minister, written by its UN ambassador and addressed to the Security Council.
Ethiopian Prime Minister Abiy Ahmed sent troops into Tigray in November to disarm and detain leaders of the region's once-dominant political party, the Tigray People's Liberation Front (TPLF).
For months the Ethiopian and Eritrean governments denied Eritreans were involved, contradicting testimony from residents, rights groups, aid workers, diplomats and even some Ethiopian civilian and military officials.
Abiy finally acknowledged the Eritreans' presence in March while speaking to lawmakers, and vowed soon after that they would leave.
Friday's letter from Eritrea said that with the TPLF "largely thwarted", Asmara and Addis Ababa "have agreed – at the highest levels – to embark on the withdrawal of the Eritrean forces and the simultaneous redeployment of Ethiopian contingents along the international boundary."
On Thursday UN aid chief Mark Lowcock told the Security Council that despite Abiy's earlier promise, there had been no evidence of a withdrawal of Eritrean troops from the region.
He also said aid workers "continue to report new atrocities which they say are being committed by Eritrean Defense Forces."
Eritrea's information minister, Yemane Gebremeskel, said on Twitter on Saturday that Asmara had summoned the UN's resident coordinator in Eritrea and the local head of the UN humanitarian coordination office to protest "wayward practices and fallacious reports... on basis of opaque networks/affiliations with TPLF."
Allegations of mass rape, massacres
Tigray residents have repeatedly accused Eritreans of mass rape and massacres, including in the towns of Axum and Dengolat.
Both Eritrea and Ethiopia blame the conflict on TPLF-orchestrated attacks on federal army camps in early November and describe it as a campaign to restore law and order.
Eritrean UN ambassador Sophia Tesfamariam reiterated this position in her letter Friday.
"We are indeed appalled by attempts to blame those who were forced to resort to legitimate measures of self–defence that other countries would have done under similar circumstances," she wrote.
"The allegations of rape and other crimes lodged against Eritrean soldiers is not just outrageous, but also a vicious attack on the culture and history of our people."
Fears of starvation
Abiy declared victory in Tigray in late November after federal forces took the regional capital Mekele, but the TPLF vowed to fight on and fighting has continued.
The conflict arrived in the middle of the harvest in Tigray and for months humanitarian access was greatly restricted, prompting fears of widespread starvation.
In his comments on Thursday the UN's Lowcock said he had received a report of 150 people dying of hunger in one area of southern Tigray, calling it "a sign of what lies ahead if more action is not taken".
Ethiopian state media on Friday night aired a report denouncing the claim as "false" and "aimed at tarnishing the image of the country."
"The humanitarian assistance being provided in the Tigray region is going well and so far, no life has been lost due to hunger," Mitiku Kassa, head of Ethiopia's national disaster commission, was quoted as saying.
Yet earlier Friday Abadi Girmay, agriculture chief of Tigray's Abiy-appointed interim administration, warned of a hunger "crisis" if farming activities don't resume.
"If we don't start getting harvest starting from this year, a very tough problem may come about that could last from three to five years," Abadi told state-affiliated Fana Broadcasting Corporate.
"If famine also sets in, a tragic history may come about that has never been seen before in this country."