They were seized by armed attackers while travelling in Ethiopia's remote northeast Afar region.
Eight Ethiopians assisting the Europeans were also taken hostage but their fate remained unclear, Beckett said.
Regional officials and Afar locals had said the hostages were marched into neighbouring Eritrea - something Eritrea had earlier denied.
Beckett said there were varying reports as to where the hostages had been held but "certainly they were released with the help of the Eritrean government".
The five Europeans were named as Peter Rudge and Jonathan Ireland, first secretary and administrative support at the British embassy in Addis Ababa, Malcolm Smart and Laure Beaufils working for the Department for International Development in the Ethiopian capital and Rosanna Moore, wife of the head of the British Council there.