The club's sporting director will take charge until the winter break after president Vincent Labrune spoke of the need for a "psychological shock" to turn around the team's form
Last season's Ligue 1 runners-up currently sit 10 points adrift of league leaders Paris Saint-Germain following defeat to Nantes on Friday, while they have lost all five of their Champions League group games so far this season.
Marseille have confirmed on their official website that Baup has been relieved of duties, with Anigo set to take charge of the team until January.
"Elie Baup is widely appreciated for his human qualities. Nobody can forget the work he did last year," said president Vincent Labrune.
"But this season, he has not obtained the expected results. A psychological shock must take place to allow us to reach our ambitions. The team is not displaying the desire and quality you would expect.
"As is unfortunately the case in these instances, the coach pays the price for the poor performances, but now I expect a very quick response from my players."
Anigo will hold a press conference with Labrune on Monday and will be on the bench for l'OM's Champions League clash with Borussia Dortmund on Wednesday.