The 19 were freed following mediation by South Sudan's government, it adds.
This is the first group to be released since about 125 children were abducted in the cross-border raid.
Members of the Murle community carried out the attack. They have previously been accused of stealing livestock and children to raise as their own.
More than 200 people from the rival Nuer community in Ethiopia's western Gambella province were killed in the 15 April raid.
Ethiopian forces crossed into South Sudan, encircling villages where the children were held.
However, the children were freed without any fighting after South Sudanese officials entered into negotiations with the abductors, the Ethiopian News Agency reports.
Negotiations would continue to free all the children, it reports.
"The children must be rescued and be reunited with their families. The cattle that was taken should also be handed over to the right owners," South Sudan's ambassador to Ethiopia, James Pitia Morgan, is quoted as saying.
Ethiopia shares a long border with South Sudan and cross-border raids involving the Murle and Nuer communities are not uncommon.
However, the scale of the 15 April shocked many people in both countries, and led to protests in Ethiopia's Gambella region with parents demanding greater protection for their children.
Ethiopia's Prime Minister Hailemariam Desalegn said "primitive and destructive forces" carried out the raid.
At least 19 Ethiopian children held captive by an armed South Sudanese group for nearly a month have been freed, Ethiopian state media reports.