Colman described filming The Favourite as "my most favourite time ever" and joked that she would scratch the names of her co-stars, Weisz and Emma Stone, onto the trophy.
The film was 20 years in the making and shone a light on a lesser-known period of royal history. Colman stars as the eccentric monarch, with Weisz as the queen's confidante and Stone as an ambitious newcomer determined to inveigle her way into Anne's affections.
Sandy Powell, who won for her costume design, said the film was "quite bonkers".
The Bafta success is a boost for The Favourite's Oscar hopes.
It will face strong competition at the Academy Awards from Roma, the black-and-white, Spanish language drama which also won the Bafta for best cinematography.
A Star is Born took home only one prize, for best original music.
Rami Malek won the leading actor award for his performance as Freddie Mercury in Bohemian Rhapsody.
The American said: "I thank my lucky stars. It's quite difficult entering this role as an outsider — you Brits do it so well and it's not lost on me how sacred your musical heritage is."
Richard E. Grant, who has been running an enthusiastic social media campaign this awards season after being nominated for Can You Ever Forgive Me?, lost out in the best supporting actor category to Mahershala Ali for Green Book.
The short film award went to 73 Cows, the story of a vegetarian farmer from Derbyshire who sent his beef herd to an animal sanctuary instead of the slaughterhouse.
Letitia Wright, winner of the EE Rising Star award, made the most emotional speech of the night as she disclosed she had been suffering from depression.
Producers Stephen Woolley and Elizabeth Karlsen, the founders of independent film company Number 9 Films, received the outstanding British contribution to cinema award for a career that has included Absolute Beginners, Mona Lisa and Carol.
Joanna Lumley, hosting the ceremony for the second year running, poked fun at the A-list guests for "battling through the weather in a chauffeur-driven limousine" to reach the Royal Albert Hall.
They included the Duke and Duchess of Cambridge, who attended in the Duke's capacity as president of Bafta. He presented the Bafta Fellowship to Thelma Schoonmaker, film editor to Martin Scorsese for more than 50 years.
Schoonmaker disclosed afterwards that the Duke had sent a message before the ceremony checking how he should say her name.
"It was very sweet of him to make sure he pronounced it properly," she said.
Lumley's opening monologue featured a joke about BlacKkKlansman, Spike Lee's drama about an African-American detective who infiltrates the white supremacist movement, showing at the "Klan Film Festival". The joke was met with a smattering of nervous laughter.
- Telegraph Media Group
