The visit by the team, which, according to a Commission spokesman in Brussels, will discuss "all subjects, no matter how sensitive", will take place at the start of next week, in good time to submit a report to EU foreign ministers on 26 January.
With Britain in the EU chair, the troika mission will be headed by Derek Fatchett, Minister of State at the Foreign Office, accompanied by his opposite numbers from Luxembourg and Austria, thus substituting junior ministers for senior officials rejected by Algiers on Wednesday.
The change of mind does spare Robin Cook what would have been a notable embarrassment just a fortnight into Britain's six-month European Presidency. Indeed, the Foreign Secretary himself achieved the breakthrough, in a telephone conversation with his Algerian opposite number, Ahmed Attaf.
But whether the visit does anything to hasten an end to the carnage which has taken some 75,000 lives in the past six years, and up to 1,700 during the first 16 days of the current holy month of Ramadan alone, is another matter.
Yesterday did bring other fragile grounds for hope, with the release of a leading member of the outlawed FIS Islamic fundamentalist group, permission for the British Ambassador, Francois Gordon, to visit Sidi Hammed, south-west of the capital where at least 100 people were slaughtered by rebel guerrillas on Sunday, and apparent approval for a separate visit by members of the European Parliament in early February.
In Algiers, Abdelkader Hachani, third ranking leader of the FIS, was freed after being detained for 24 hours for giving interviews to two French newspapers, in which he urged foreign countries to press the Algerian government to open talks with his group. Technically, Mr Hachani could still be tried for his breach of a three-year ban on such media contact, but even if it proves merely temporary, his release seems timed to improve the political climate during the EU visit.
In Sidi Hammed, Mr Gordon was told by survivors that they had sought guns in vain from the Algerian authorities with which to protect themselves.
Another survivor, whose wife was murdered, said he tried to call the security forces during the attack, but they did not come in time. This is precisely the sort of allegation which has fanned rumours of tacit government connivance in some of the massacres, a charge furiously denied by Algiers, as it resists foreign involvement. "It is up to the English and Americans to come and protect us," the man said.