The film also won the Sound prize.
Cuaron thanked Netflix for getting behind the film, which is in Spanish and has garnered 10 Oscar nominations, including for Best Picture, a major recognition for the streaming service.
The prestigious United Kingdom academy named black-and-white film "Roma" best picture of 2018, while also honoring its director Alfonso Cuaron as best director.
Picking up her BAFTA, Rachel Weisz paid tribute to her co-stars, saying she had "the greatest luck in that I played opposite two of the most glorious women" while Olivia Colman said in her acceptance speech that her fellow actresses in the film had been the "best and classiest and coolest honour guard any woman could ever have" before adding "as far as I'm concerned, all three of us are the same....this is for all three of us".
It ended the night with a total seven awards, including a supporting actress accolade for Rachel Weisz and a leading actress honour for Olivia Colman.
Lady Gaga, Lukas Nelson and Bradley Cooper were recognised in the Original Music category for A Star Is Born soundtrack.
"Didn´t we have an extraordinary time?"
Weisz and Stone are also in the running for Oscars in a proud year for the British film industry.
Mahershala Ali took the Supporting Actor prize for his role in Green Book, set in the segregated US South in the early 1960s.
It was the big night before the Oscars with everyone waiting to see who would win big at the Royal Albert Hall.
"Free Solo" - a film about the first person to climb El Capitan in Yosemite National Park without ropes - won the award for best documentary.
Last year´s ceremony at the Royal Albert Hall saw Hollywood stars paint the red carpet black in solidarity with the #MeToo movement. The period drama also went home with the gong for most outstanding British film, beating a field of contenders that included the box office hit Bohemian Rhapsody.
This referred to an article in the U.S. magazine The Atlantic detailing sexual misconduct allegations that included underage sex.
Singer denies the story and the film itself was still up for the award, although it failed to win it.
"If they do good movies, they should be recognised".
Meanwhile, Rami Malek won the Leading Actor prize for his role playing late Queen frontman Freddie Mercury in Bohemian Rhapsody.