The movie about how Britain's King George VI overcame a debilitating stammer on the eve of World War II has a cast that includes Guy Pearce, Helena Bonham-Carter, Michael Gambon and Geoffrey Rush.
Colin Firth, who played the speech-impaired monarch, also picked up the SAG prize for Best Male Actor in a Leading Role in a Film.
"This is really something," Firth said. "Until today, I would say, probably, if ever I had a trophy that told me something's really happening for me it was my SAG card, to be a part of this extraordinary collective.
"Growing up in England, it's not something you expect to see in your wallet and so it has this glow. I used to flash it around, hoping it would get me female attention, and entry into nightclubs and top-level government departments and it didn't.
"Now I've got this (award,) I'd like to thank -- looking at who's here -- I'd like to thank security for letting me into the building."
Natalie Portman won the SAG prize for Best Female Actor in a Leading Role in a Film for her work in the ballet drama "Black Swan."
"I've been working since I was 11 years old and SAG has taken care of me. They made sure that I wasn't working too long and that I got my education while I was working and I'm so grateful to have this union protecting me every day," Portman said.
"The Fighter" co-stars Melissa Leo and Christian Bale won the SAG Awards for Best Female and Male Actor in a Supporting Role.