Anigo took the helm at the Stade Velodrome earlier in the month following the dismissal of Elie Baup.
L'OM failed to register a single point in Champions League Group F, losing home and away to Arsenal, Borussia Dortmund and Napoli.
The nine-times French champions are currently sixth in the top flight, 15 points adrift of leaders Paris St Germain, and defeat to Nantes earlier in month proved to be the last straw for Baup.
Despite making an early exit from Europe and struggling to keep pace in Ligue 1, Labrune believes Anigo is the man to weather the storm.
Furthermore the club president suggested Anigo could feature in the long-term plans at the Stade Velodrome.
Speaking to L'Equipe, he said: "Jose will be present when training sessions resume in January.
"He will go on as long as needed. My short-term priority is restoration of the team, both in minds and on the pitch.
"For such purpose, Jose is the right man in the right place. He is the one who knows the group best as he hired all the players.
"That solution is not incompatible with our long term strategy, set last summer. The current cyclical situation won't change anything."
Marseille host fourth-placed Bordeaux on Sunday with the opportunity to leapfrog their opponents in their final game before the winter break.