"It was my hardest decision as club president," president Vincent Labrune said on the club's website.
Marseille lost 1-0 at home to Nantes on Friday and are fifth in the standings with 27 points from 17 games.
They are 10 points adrift of arch rivals and leaders Paris St Germain, who play their game in hand at home against Sochaux on Saturday.
Baup had been appointed last year and took the club to second in Ligue 1 at the end of the 2012-13 season.
Marseille, however, have scored no points in the Champions League this season after five games in the group stage.
Anigo, 52, led OM to the UEFA Cup final in 2004 after he had replaced Alain Perrin in January. The club said he will be in charge until the winter break, which starts on Dec. 23.