The win for best drama solidified its prospects as an Academy Awards favorite.
Colin Firth won best dramatic actor for his role in the 1930s British monarch saga, "The King's Speech" and Natalie Portman won best dramatic actress as a ballerina in the psychosexual thriller "Black Swan."
Firth, however, wasn't looking too far into the future.
"Oh, the Oscar question," he said. "I basically am just about able to project myself to the end of this press conference. I can't get further than that in my mind."
Lead-acting honors for the Globes' musical or comedy categories went to Annette Bening for the lesbian-family story "The Kids Are All Right" and Paul Giamatti for the curmudgeon tale "Barney's Version."
"The Fighter," also a contender for best drama, earned both supporting actor Globes, for Christian Bale and Melissa Leo.
Aaron Sorkin, screenplay winner for "The Social Network," had kind words for Facebook founder Mark Zuckerberg, played by Jesse Eisenberg.
"Mark Zuckerberg, if you're watching, Rooney Mara makes a prediction at the beginning of the movie. She was wrong. You turned out to be a great entrepreneur, a great visionary and an incredible altruist," Sorkin said.
Robert De Niro received the Cecil B. DeMille Award for career achievement.
Sunday's ceremony, hosted by Ricky Gervais for the second-straight year, was held at the Beverly Hilton Hotel.
Nominations voting for the Feb. 27 Oscars closed Friday, and nominees will be announced Jan. 25.
The Associated Press contributed to this report.