The dark comedy had led the nominations for the annual ceremony with nods in four categories but picked up just one, the most coveted accolade of the night, Outstanding Motion Picture Cast.
Accepting the award, Naomi Watts tripped over co-star Emma Stone's dress but was saved from falling by Edward Norton and regained her composure to give her thanks for such a "wonderful, collaborative" experience.
And Michael Keaton said: "Thank you, this is very, very exciting. It's the ultimate team sport, what we do for a living. To look at all the great actors out there...
"These guys are tremendous. I'm proud to be a part of this group."
British star Eddie picked up Outstanding Male Actor in a Leading Role for his portrayal of Stephen Hawking in 'The Theory of Everything' - for which he has already won a Golden Globe award - and insisted the honour was "unexpected".
He said: "It was really unexpected. There's really a lot of euphoric noise going on in my head."
During his acceptance speech, Eddie, 33, paid tribute to his fellow nominees, including 'Birdman' star Michael, and dedicated the win to the acclaimed physics professor and his ex-wife, as well as sufferers of motor neuron disease - which is also known as ALS (amyotrophic lateral sclerosis) - around the world.
He said: "I'd like to dedicate the award, this very wonderful, skinny man, to those people around the world living with ALS, to those who have lost their lives to this brutal disease
"In preparing to play Stephen, I met many people fighting ALS and I met their families fighting alongside them and the courage and the bravery and the spirit -- the triumphant spirit I witnessed -- blew my mind. So I'm incredibly grateful to them.
"To Stephen and Jane back in Cambridge, thank you for allowing someone who gave up science when they were 14 years old to enter your orbit...Thank you for reminding me of the overwhelming power of the will to love and the will to live every second of your life as fully and as passionately as possible."
Meanwhile, 'Orange is the New Black' picked up the Outstanding TV ensemble prize, while its star Uzo Aduba picked up the Outstanding Female Actor in a Comedy Series accolade and made an emotional speech.
She said: "Oh my God. I really, truly do not know what to say.
"I just wanna say thank you so much to the Screen Actors Guild for this incredible honour. I want to thank Jenji Kohan for writing a show like this and putting something like this on television, not just for myself but for our incredible team of actors to be seen in such a beautiful way.
"I want to say a huge, massive thanks to our cast, without whom this is not possible.
"I love you guys so desperately, so much. This is not done without you at all, any step of the way. I want to thank my friends. I want to thank my family and just finally I really, really, truly want to say that the day I got this job was the day I had stopped acting, and to be in a room with all you amazing human beings, amazing talents, for what I love and respect so much, is really, truly the greatest honour. Thank you so very much. God bless you."
Other winners included 'Still Alice' star Julianne Moore, who picked up Outstanding Female Actor in a Leading Role, Kevin Spacey, who scooped Outstanding Male Actor in a Drama Series for his work on 'House of Cards', and 'Downton Abbey', whose cast were honoured with Outstanding Ensemble in a Drama Series.
Carrie Fisher presented a Lifetime Achievement award to her mother Debbie Reynolds.
21st Screen Actors Guild (SAG) Awards list of winners:
Outstanding Female Actor in a Comedy Series:
Uzo Aduba, 'Orange Is the New Black'
Outstanding Male Actor in a Comedy Series:
William H. Macy, 'Shameless'
Outstanding Ensemble in a Comedy Series:
'Orange Is the New Black'
Outstanding Male Actor in a TV Movie or Miniseries:
Mark Ruffalo, 'The Normal Heart'
Outstanding Female Actor in a TV Movie or Miniseries: Frances McDormand, 'Olive Kitteridge'
Outstanding Male Actor in a Drama Series:
Kevin Spacey, 'House of Cards'
Outstanding Female Actor in a Drama Series: Viola Davis, 'How to Get Away With Murder'
Outstanding Ensemble in a Drama Series: 'Downton Abbey'
Outstanding Female Actor in a Supporting Role:
Patricia Arquette, 'Boyhood'
Outstanding Male Actor in a Supporting Role:
J.K. Simmons, 'Whiplash'
Outstanding Male Actor in a Leading Role:
Eddie Redmayne, 'The Theory of Everything'
Outstanding Female Actor in a Leading Role:
Julianne Moore, 'Still Alice'
Outstanding Ensemble in a Motion Picture: