The wins bolster the movie's claim to being a front-runner for Academy Awards after already sweeping best picture honors from several other critics groups including those in New York, Los Angeles and Boston.
The National Society of Film critics also named Jesse Eisenberg best actor in the role of Facebook founder Mark Zuckerberg, David Fincher best director, and writer Aaron Sorkin claimed best screenplay.
Giovanna Mezzogiorno was named best actress for "Vincere," in which she plays Mussolini's lover during his early years.
"Social Network" is a fictionalized story telling of the rise of social networking website Facebook from an idea dreamed up in a college dormitory to its first one million users.
Critics' awards are important in helping build momentum heading toward the Academy Awards, or Oscars, which are the world's top film awards given out on the final Sunday in February by the Academy of Motion Picture Arts and Sciences.
The NFSC includes 61 members from major newspapers in Los Angeles, Boston, New York and Chicago as well as from Time, Newsweek and The New Yorker.
The group's awards for best supporting performances went to Geoffrey Rush, who plays King George VI's speech therapist in "The King's Speech," and Olivia Williams for the Roman Polanski thriller "The Ghost Writer."
The award for best foreign language film was won by the French-German film "Carlos," about the life of Ilich Ramirez Sanchez, a Venezuelan-born leftist revolutionary known as the Jackal who raided the OPEC headquarters in Vienna, Austria in 1975.
The film critics, in their 45th annual awards, named "Inside Job," the documentary about the 2008 financial crisis, as the year's best nonfiction film, while "True Grit" took the prize for cinematography.
Highly-touted films including "The Fighter," "Black Swan" and "The Kids Are All Right" were shut out of the awards.