The film also won awards for adapted screenplay, music (Indian composer A.R. Rahman), editing, cinematography and sound, as it continued its improbable march to the Academy Awards later this month.
"David Lean said you should announce your ambition for the film in the first five minutes, " Boyle said.
"In that spirit, I'd like to announce my ambition for this speech - my father's power went out last night. A big shout-out to everyone who helped him run the extension cable in so he could watch this."
"Slumdog Millionaire" is considered the front runner for the Oscars with 10 nominations and has already dominated the Golden Globes, while also picking up prizes from the Producers Guild of America, Directors Guild of America, Screen Actors Guild and Writers Guild of America.
The best actress award went to Kate Winslet for her role in "The Reader". Winslet, who was also nominated in the same category for "Revolutionary Road", said: "To get this award at home means a lot to me." It's her second BAFTA, the first in 1995 for her supporting role in "Sense and Sensibility".
Mickey Rourke won the best actor prize for his portrayal of a comeback fighter in "The Wrestler" and thanked director Darren Aronofsky for giving him a second chance. "Such a pleasure to be here and be out of the darkness, " he said.
Penelope Cruz took home the best supporting actress prize for "Vicky Cristina Barcelona", while the supporting actor award went posthumously to Heath Ledger for "The Dark Knight".
The "Curious Case of Benjamin Button", the Brad Pitt-starrer about a man who ages backwards, was nominated for 11, but won just three awards - for production design, special effects and hair and make-up.
"I've Loved You So Long" (France) won the top prize for best foreign film.
The ceremony of the British Academy of Film and Television Arts, held weeks before the Academy Awards in Los Angeles, is one of the highlights of the year's awards season.