The movie, which took 12 years to make, won best film drama, with Linklater named best director and Patricia Arquette best supporting actress.
British actor Eddie Redmayne won best actor in a drama for his role as physicist Professor Stephen Hawking in The Theory of Everything.
British actresses Joanne Froggatt and Ruth Wilson also collected prizes.
The awards, which cover both film and TV, saw Froggatt win best supporting actress in a TV series for her role as the maid Anna Bates in Downton Abbey.
Wilson was named best actress in a US TV drama for The Affair.
Collecting the best director award, Linklater said: “I want to dedicate this to parents that are evolving everywhere and families that are just passing through this world and doing their best.
“Bottom line is we’re all flawed in this world. No one’s perfect.”
Meanwhile, Redmayne beat competition from fellow Brits Benedict Cumberbatch (for The Imitation Game) and David Oyelowo (for Selma) to the best actor award.
On collecting his trophy, Redmayne revealed he cut short his honeymoon to attend the ceremony in Beverly Hills. He said: “I promised her sunshine – it wasn’t so great today – but thanks for giving us a honeymoon we’ll remember.”
The actor was the only Briton to win an award in the film categories despite a strong showing in the nominations.
His co-star Felicity Jones missed out on the best drama actress award to Julianne Moore, who played an academic with early onset Alzheimer’s in Still Alice.
Comedy drama Birdman had led nominations going into the ceremony with seven nods. The film collected two awards – one for Michael Keaton as best actor in a comedy or musical and one for best screenplay.
An emotional Keaton said: “You have no idea what this means to me. I want to thank the Hollywood Foreign Press for even having a comedy category.”
Whiplash star JK Simmons collected the prize for best supporting actor. He thanked his co-star Miles Teller for being a person he wanted to “slap in the face”, referencing a memorable scene in the film.
In a surprise win, Wes Anderson’s The Grand Budapest Hotel was named best comedy or musical film, with Amy Adams named best actress in a comedy or musical for her role in Big Eyes.
In the TV categories, Froggatt was surprised to win her award, beating the likes of Kathy Bates, Allison Janney, Michelle Monaghan and Uzo Aduba from Orange is the New Black.
She was rewarded for her portrayal of Anna Bates, who was raped and then accused of murdering her attacker in the drama.
“This is the most shocking moment of my life,” she said. “After the storyline aired I received a lot of letters from survivors of rape – thank you to everyone who wrote – I heard you.”
Maggie Gyllenhaal won best actress in a TV movie or mini-series for her role in BBC Two drama The Honourable Woman.
Kevin Spacey picked up his first Golden Globe after eight nominations for his role in Netflix drama House of Cards.
Amazon comedy-drama Transparent celebrated wins for best TV comedy and best actor in a TV comedy for Jeffrey Tambor’s role as a transgender woman.
Fargo also collected two prizes, winning best TV miniseries and best actor in a TV miniseries for Billy Bob Thornton.
The Affair was named best TV drama, while other awards went to Gina Rodriguez for comedy series Jane the Virgin, and Matt Bomer for Aids drama The Normal Heart.