Foreign Minister Gabi Ashkenazi worked with his counterpart in the region to secure the release.
Tigray forces fired rockets into Eritrea at the weekend.
Sigd is being observed in a scaled-down fashion this year.
The government blamed the ruling party in Tigray, the Tigray People's Liberation Front.
"There is a risk this situation will spiral totally out of control, leading to heavy casualties and destruction, as well as mass displacement within Ethiopia itself and across borders," she said.
About 7,000 of those crossing have arrived at Hamdayat in Sudan's Kassala state, with another 4,000 arriving at Luqdi in al-Qadarif state.
I dedicate my column this week to the traditional foods eaten by the Ethiopian Jewish community to celebrate Sigd, as well other dishes that are eaten on a daily basis.
Air strikes and ground combat have killed hundreds, sent refugees flooding into Sudan, stirred Ethiopia's ethnic divisions and raised questions over the credentials Prime Minister Abiy Ahmed.
This is so stupid and reckless that it makes American politics look positively demure by comparison.