"This film took 20 years to make, I contributed to the last 10," said Lanthimos upon winning the Outstanding British Film award, reported variety.com.
The film then scored its second win in the production design category.
Weisz beat Emma Stone, her own co-star from "The Favourite", to win the Best Supporting Actress.
"I had the most extraordinary luck in that I played opposite two of the most glorious women Olivia Colman and Emma Stone," said Weisz, who has bagged her first BAFTA.
Among other early winners included "Spider-Man: Into the Spider-Verse" (Animated Film), "Free Solo" (Best Documentary), "Bohemian Rhapsody" (Sound), "Vice" (Editing), "A Star Is Born" (Best Original Music) and "BlacKkKlansman" (Best Adapted Screenplay).
British-Guyanese actress Letitia Wright won the EE Rising Star Award, the only award voted for by the public, variety.com reported.
"I want to thank everybody who gave me a chance," said Wright.
Saying she had previously considered giving up acting but that her faith had helped her through, she encouraged "anybody that is going through a tough time" to know that "God made you and loves you. Let your light shine".
The ceremony at the Royal Albert Hall is being hosted for the second consecutive year by Joanna Lumley, who mocked the Oscars with her opening joke.
"Thank goodness BAFTA actually has a host. But I suspect that may h ave something do to with the fact I'm not on Twitter," said Lumley, taking a dig at how a series of old homophobic tweets resulted in comedian Kevin Hart to step down as host of the 2019 Oscars.