Also nominated for the honor were the ensembles of "The Curious Case of Benjamin Button," "Doubt," "Frost/Nixon" and "Milk."
"It was overwhelming enough to be nominated, but, to win this, is unbelievable," said "Slumdog" co-star Anil Kapoor.
"Milk" star Sean Penn picked up the award for best actor in a film, beating out Richard Jenkins of "The Visitor," Frank Langella of "Frost/Nixon," Brad Pitt of "Benjamin Button" and Mickey Rourke of "The Wrestler" for the title.
"As actors, we don't play gay, straight ... We play human beings," Penn reminded the audience, referring to, Harvey Milk, the real-life gay rights activist he played. "I and we are all so appreciative for this kind of acknowledgment because this is a story about equal rights for all human beings."
Tearing up, Penn addressed his fellow best actor nominees: "I wept at everyone of these guys' performances. You're stunning."
Running up to the stage to collect her prize, Streep breathlessly said, "Thank you."
Gesturing toward her black pants outfit, she exclaimed with a laugh: "Well, I didn't even buy a dress. I'm really, really shocked and, even though awards mean nothing to me any more, I'm really happy."
Also nominated for the best actress in a film award were Hathaway for "Rachel Getting Married," Winslet for "Revolutionary Road," Angelina Jolie for "Changeling" and Melissa Leo for "Frozen River."
Winslet won the best supporting actress prize for "The Reader" and the late Heath Ledger earned the trophy for "The Dark Knight."
2014: The Year in Music [PHOTOS]