The Iran hostage drama directed and starring Affleck, won best cast ensemble, the top prize, at the awards in Los Angeles last night.
Steven Spielberg's top-tipped presidential drama Lincoln also scored successes with best actor for Daniel Day-Lewis in the title role, and supporting actor Tommy Lee Jones.
British actor Day-Lewis, who also won the best actor Golden Globe for his role as America's 16th president, joked about having hesitated before accepting the role.
He said: "It occurred to me it was an actor that murdered Abraham Lincoln.
"And therefore it's only fitting that every now and then an actor tries to bring him back to life."
Hunger Games star Jennifer Lawrence, 22, took best actress award for her role in romantic comedy Silver Linings Playbook, over Jessica Chastain's CIA agent in the Osama bin Laden thriller Zero Dark Thirty.
Jones, 66, won the best supporting actor trophy for his turn as radical congress-man Thaddeus Stevens in Lincoln, beating strong competition from Robert De Niro, who played a gruff father in Silver Linings Playbook.
Anne Hathaway, 30, won her first Screen Actor's Guild (SAG) award for her support-ing role as tragic Fantine in the musical Les Miserables.
Accepting her award, she said: "I got my SAG card when I was 14, and I have loved every single minute of my life as an actor."
On the small screen, Downton Abbey won best ensemble cast.
Actress Phyllis Logan –who was born in Johnstone, Renfrewshire – and who plays Mrs Hughes in the show, accepted the award.
She said the cast, including co-star Michelle Dockery who was also at the event – were "over-whelmed" by the achievement.
Modern Family won the best comedy cast ensemble award for a third consecutive time, while the award for best television drama actor went to Bryan Cranston for Breaking Bad. Claire Danes won best drama actress for Homeland.
Alec Baldwin won best TV comedy actor for the eighth time for his role as an egotist-ical executive in 30 Rock and his co-star Tina Fey took the honours for comedy actress.
Hollywood legend Dick Van Dyke was awarded the SAG lifetime achievement award.
The 87-year-old said: "They tell me you never work again once you get this award.
"I'll have to let them know I'm available."