Anelka was sent home after a verbal assault on France coach Raymond Domenech and will play no further part in the tournament, although Les Bleus have only a slim chance of getting beyond the group stage in any case after a 0-0 draw with Uruguay and a 2-0 defeat by Mexico.
Drogba, quoted on news24.com in South Africa, said: "He has been a team-mate for two and a half years. Yes, I support him.
"I wish him good luck, it's a difficult time for him. He is very strong mentally, he's a competitor. I was surprised he was kicked out of the team."
The France squad refused to train on Sunday in support of Anelka. They play their final Group A match against World Cup hosts South Africa in Bloemfontein on Tuesday.
The president of France Nicolas Sarkozy, meanwhile, has asked his sports minister Roselyne Bachelot to meet with the key people involved in the bitter national team row which has wrecked their World Cup chances.
She told TFI television: "We are taking note of the indignation of the French people and.calling for dignity and responsibility.
"It's not yet the right time to take disciplinary action but that time will come very soon."
Economy minister Christine Lagarde, who represented her country as a synchronised swimmer, told LCI television: "I am appalled. I am appalled because I have worn the French national colours as you know, and when you wear the French national colours you have added responsibilities.
"The first (responsibility) is to be the best possible in sport, but also to be exemplary because you are watched. There are young children, young adolescents who practise the sport and look to you as an example."