The Favourite won seven trophies, including best British film and best actress for Olivia Colman, who plays Britain's 18th century Queen Anne in the female-centric drama.
Alfonso Cuaron's Roma, which centres on the nanny to a middle-class Mexico City family, took out the prizes for best picture, director, cinematography and foreign-language film.
Cuaron thanked Netflix for backing "a black-and-white film about a domestic worker'' that is not in English.
It bested The Favourite, which bookies had favoured to win the top prize at Britain's equivalent of the Oscars.
Director Yorgos Lanthimos' royal film snapped up the outstanding British film and screenplay awards, as well as prizes for its opulent production design, its extravagant costumes, larger-than-life hair and makeup and the performances of Colman and supporting actress Rachel Weisz.
"This is for all three of us," Colman said, speaking of Weisz and the film's other star, Emma Stone. "It's got my name on it, but we can scratch on some other ones."
The British actress had been favoured to win the award for her performance as Britain's 18th-century Queen Anne in the royal tragicomedy.
A delighted Colman said: "We're doing so well, aren't we? We're going to get so pissed later.''
Supporting actress winner Weisz played Sarah Churchill, who was one of the most powerful people in 18th-century England.
The actress said she "had the greatest luck in that I played opposite two of the most glorious women", referring to co-stars Emma Stone and Olivia Colman.
Rami Malek won the best-actor prize for his performance as Freddie Mercury in Bohemian Rhapsody.
Malek said it was "extraordinary'' to be recognised for his performance as the late frontman of Queen.
Mahershala Ali was named best supporting actor for his role as a concert pianist in Green Book.
Ali said it was "surreal to get this kind of thing for something you feel so passionate about and grateful for".
Other winners include Spike Lee's BlacKkKlansman for best adapted screenplay and the Bradley Cooper-directed A Star is Born for best original music.
The Bafta results will be scoured for clues on who might triumph at Hollywood's Academy Awards on February 24. Roma and The Favourite each have 10 Oscar nominations.