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The Trial of the Chicago 7, set during a Vietnam War protest in 1968, took the prize for best ensemble with a cast that includes Sacha Baron Cohen, Eddie Redmayne, Frank Langella and Mark Rylance.
Langella credited writer and director Aaron Sorkin for the win, saying his "voice is the soul of this movie."
All four of the winning film actors were people of colour, the first time this has happened in the SAG awards' 27-history.
Chadwick Boseman, who died of cancer last year at age 43, picked up another award for his final role as an ambitious 1930s trumpet player in Ma Rainey's Black Bottom.
Sunday's SAG win positioned him as front-runner for his first Oscar at the Academy Awards on April 25.
Viola Davis won best actress for playing a jazz diva in Ma Rainey, besting presumed favourites Frances McDormand for Nomadland and Carey Mulligan in revenge drama Promising Young Woman.
The SAG awards, chosen by members of the acting union, are closely watched because actors form the largest voting group in the Academy of Motion Picture Arts and Sciences, which organises the Oscars.
The win for The Trial of the Chicago 7 boosted the film's chances of winning the best picture Academy Award in a year that has seen modern recession-era movie Nomadland grab many of the pre-Oscar awards.
Because of the coronavirus pandemic, this year's SAG ceremony was scaled down from a gala dinner to a one-hour presentation, held remotely, and peppered with jokes about how actors filled their free time during lockdown or got a start in their careers.
Helen Mirren spoke of a close encounter with a bear during the pandemic, Sterling K Brown demonstrated his juggling skills and Jimmy Fallon played the harmonica.
Minari, the Korean-language story of an immigrant family struggling to make it in 1980s Arkansas, brought a supporting actress award for South Korean veteran Youn Yuh-jung as a cantankerous grandmother.
"I am very pleased and happy," said 73-year-old Youn.
British actor Daniel Kaluuya won for his supporting role as the late Black Panther activist Fred Hampton in Judas and the Black Messiah.
The SAG awards also recognise television performers, with the casts of Schitt's Creek and The Crown taking the top prizes for comedy and drama, respectively.
Many of the SAG winners and nominees are also up for honours at the British BAFTA awards next week, and the Independent Spirit awards on April 22.