Denzel Washington was named best actor for Fences.
Several winners attacked Mr Trump`s ban on immigration from seven countries.
The tone was set by host Ashton Kutcher, who began the evening by greeting "everyone at home and everyone in airports that belong in my America".
"We`re in a really tricky time in our country and things are very inexcusable and scary," said Emma Stone as she won the best actress prize for La La Land.
Mahershala Ali, meanwhile, spoke about his Muslim faith as he accepted the best supporting actor prize for Moonlight.
Mr Trump`s executive order, signed on Friday, halted the entire US refugee programme for 120 days, indefinitely banned Syrian refugees, and suspended all nationals from seven Muslim-majority countries.
Lithgow and Foy were recognised for playing Winston Churchill and the Queen in The Crown
Julia Louis-Dreyfus called the immigration ban "a blemish" and "un-American" as she won best comedy TV actress for Veep.
And there were cheers for actress Taylor Schilling when she described the award-winning cast of Orange is the New Black as "a diverse group of people, representing generations of families who have sought a better life here".
Denzel Washington`s Fences co-star Viola Davis was named best supporting actress at an event where several actors of colour were honoured.
Yet the actress told reporters that every black nominee had been shortlisted "because they deserve to be" and "not... because of the colour of their skin."
Meanwhile, Lily Tomlin was presented with a lifetime achievement award by her 9 to 5 co-star Dolly Parton at Sunday`s ceremony.
There were also awards for John Lithgow and Britain`s Claire Foy for their work in The Crown, Netflix`s drama about the British royal family.
Hidden Figures` success came at the expense of such Oscar-tipped titles as Manchester by the Sea, which left empty-handed despite being up for four prizes.
Oscar favourite La La Land was not up for the outstanding performance by a cast in a motion picture award - the SAG equivalent of the best picture prize.
Compete list of winners
Outstanding male actor in a leading role: Denzel Washington, Fences
Outstanding female actor in a leading role: Emma Stone, La La Land
Cranston was honoured for playing Lyndon B Johnson in All the Way
Outstanding male actor in a supporting role: Mahershala Ali, Moonlight
Outstanding female actor in a supporting role: Viola Davis, Fences
Outstanding cast in a motion picture: Hidden Figures
Outstanding male actor in a TV movie or limited series: Bryan Cranston, All the Way
Outstanding female actor in a TV movie or limited series: Sarah Paulson, The People Versus OJ Simpson
Outstanding male actor in a drama series: John Lithgow, The Crown
Outstanding female actor in a drama series: Claire Foy, The Crown
Outstanding male actor in a comedy series: William H Macy, Shameless
Outstanding female actor in a comedy series: Julia Louis-Dreyfus, Veep
Winona Ryder (second from right) was among those honoured for sci-fi series Stranger Things
Outstanding ensemble in a drama series: Stranger Things
Outstanding ensemble in a comedy series: Orange is the New Black
Outstanding stunt ensemble in a comedy or drama series: Game of Thrones
Outstanding stunt ensemble in a motion picture: Hacksaw Ridge
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