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.