The biographical comedy-drama took the trophy for outstanding performance by a cast in a motion picture.
Starring Taraji P. Henson, Octavia Spencer and Janelle Monae, it took the trophy for outstanding performance by a cast in a motion picture, the SAG equivalent of a best film Academy Award.
Theodore Melfi's movie, which recounts the story of African-American mathematicians who helped NASA put the first men in space, was a shock winner, with Kenneth Lonergan's "Manchester by the Sea" widely tipped for the top prize.
It also beat coming-of-age drama "Moonlight," comedy "Captain Fantastic" and race relations story "Fences."
"Manchester" star Casey Affleck -- the frontrunner for the best actor Oscar -- was also shut out, with the SAG award going instead to Denzel Washington for "Fences."
Washington, 62, who has two Oscars, also beat competition from Andrew Garfield ("Hacksaw Ridge"), Viggo Mortensen ("Captain Fantastic") and Ryan Gosling ("La La Land").
In the best actress category, Emma Stone won for "La La Land," Damien Chazelle's whimsical tribute to the musicals of Hollywood's golden age.
The support acting awards went to Mahershala Ali for "Moonlight" -- his second trophy of the night after the "Hidden Figures" cast prize -- and Viola Davis, who made it a double for "Fences."
Ali, 42 -- best known for his portrayal of White House chief of staff Remy Danton in Netflix political drama "House of Cards" -- gave an emotional speech about the need to support the marginalized.
"I think what I've learnt from working on 'Moonlight' is we see what happens when we persecute people -- they fold into themselves," said Ali, a Muslim with a Christian minister for a mother, fighting back tears.
'This award is legitimate'
On the television side, "The Crown," which follows the lives and loves of Britain's post-war Royal Family, picked up best actor awards for John Lithgow and Claire Foy.
Netflix's nostalgic breakout hit "Stranger Things" -- about a gang of 1980s children looking for their missing friend -- won the award for best cast in a drama series.
The SAG nominations are the second major announcement in Tinseltown's glittering awards season, which climaxes with the Academy Awards on February 26.
The evening was marked by speeches denouncing President Donald Trump's immigration ban targeting several Muslim-majority countries.
Julia Louis-Dreyfus followed suit as she picked up her second SAG award for comedy acting in "Veep" -- her eighth statuette overall.
In a speech aimed solely at Trump, Louis-Dreyfus -- whose father fled Nazi-occupied France -- said she loved the US but hated its blemishes, adding "this immigration ban is a blemish."
The 56-year-old also poked fun at Trump's focus on attendance at his inauguration and allegations of voter fraud.
"Whether the Russians did or did not hack the SAG awards, I look out across the million, maybe 1.5 million people in this room and say this award is legitimate and I won it," she said.
"La La Land", set in modern-day Los Angeles, cleaned up at the Golden Globes earlier in January, winning all seven categories in which it was nominated.
Although less high-profile than the Globes, the SAG awards are seen as more of an indicator of Oscars glory, since members represent some 1,200 of the roughly 6,000 voters for the Academy of Motion Picture Arts and Sciences.
Last year, "Spotlight," a drama about sexual abuse in the Catholic church, took the prize for outstanding performance by a cast in a motion picture -- and went on to win the best film Oscar.
The awards season is being closely watched as it follows controversy over the lack of diversity in last year's list of Oscar nominees.
The scandal was largely stopped in its tracks this year with a high proportion of black actors among the nominees for February's Oscars.
The SAG awards escaped criticism in any case with a good number of ethnic minority wins last year.
Washington was the only black nominee among the 10 hopefuls for best lead role in a movie, although there were four nominees from ethnic minority backgrounds in the supporting categories.
Sterling K Brown, Courtney B Vance, Riz Ahmed, Rami Malek, Thandie Newton, Audra McDonald and Kerry Washington were all nominated for individual TV acting awards, although they all lost out to white actors.
"We've been through more difficult times. We got through it then during the segregation era and we'll get through it now," Janelle Monae of "Hidden Figures" said backstage.
"As the great Kevin Costner said, we all pee the same color."