Redmayne said it was “one of the best nights of my life”.
The film was also named outstanding British film and won a third award for its adapted screenplay.
Coming of age drama Boyhood – shot over 12 years with the same cast – was named best film, with Richard Linklater picking up best director prize.
Patricia Arquette also won the best supporting actress Bafta for her role in the film.
The ceremony at London’s Royal Opera House on Sunday night was hosted by Stephen Fry.
Julianne Moore won the leading actress prize for her performance as a linguistics professor with early-onset Alzheimer’s.
Collecting her award, she thanked “everybody in the Alzheimer’s community who were so generous with their time and telling me their experiences”.
Wes Anderson’s quirky comedy The Grand Budapest Hotel won the most awards on the night – a total of five including costume design, production design, make-up and original music; with Anderson winning his first Bafta for original screenplay.
Jazz drumming drama Whiplash also took three awards – for editing, sound and supporting actor for JK Simmons, who thanked director Damien Chazelle for “the gift of this character”.
World War Two drama The Imitation Game, with Benedict Cumberbatch as codebreaker Alan Turing, won nothing despite its nine nominations.
Redmayne, who was the favourite to win best actor award, arrived on stage to roars of approval from the star-packed audience.
He said of the Hawking family: “I want to thank them for their trust, their generosity and their kindness.
“And for reminding me of the great strength that comes from having the will to live a full and passionate life.”
The Theory of Everything won the first award of the night – for outstanding British film – presented by David Beckham.
Producer Eric Fellner gave “heartfelt thanks” to Professor Hawking and Jane Wilde Hawking, on whose book the film was based: “I hope you all feel we’ve done you proud.”
Prof Hawking had earlier appeared on stage to present the award for special visual effects, alongside leading actress nominee Felicity Jones, which went to Christopher Nolan ‘s sci-fi epic Interstellar.
Jones joked she was with the “only person on the planet more intelligent than Stephen Fry”.
Prof Hawking responded: “Yes, and better looking.”
This year’s Baftas didn’t deliver any major surprises but they do set up an an exciting battle for the Oscars in two weeks’ time.
Will Boyhood repeat its British success and go on to take best picture in Hollywood?
For the past six years the best film winner chosen by British Academy voters has gone on to win an Oscar. Last year it was 12 Years a Slave, and before that The Artist, The King’s Speech, The Hurt Locker and Slumdog Millionaire.
But the Baftas are international awards with a local twist and only some 20% of the US Academy is made up of Brits.
Birdman may have only won a single Bafta for cinematography, but this weekend it took the top prize at the Director Guild Awards in the US, which have proved a very reliable indicator of Oscars success.
Boyhood star Ellar Coltrane – who was cast in the film when he was aged six – was among the cast and crew who picked up the best film award from actor Tom Cruise.
“The truth is, it didn’t feel like a movie, more like an exercise in collaboration and vulnerability,” he said.
“To have this movie recognised alongside such grand pieces of art means life itself must be more exciting than we let on.”
Best supporting actress winner Patricia Arquette thanked Boyhood director Linklater, saying: “You broke the rules of cinema… and you made an ordinary story extraordinary.”
Showbiz satire Birdman, staring Michael Keaton as a washed up superhero actor, took only one award – for Emmanuel Lubezki’s cinematography.
Lubezki won the same prize last year for space thriller Gravity.
Polish film Ida – centred around a young woman training to be a nun – won the award for film not in the English language.
The outstanding debut award went to Stephen Beresford and David Livingstone for British drama Pride, the true story of gay and lesbian activists who supported striking miners in the 1980s.
Beresford said it took him 20 years to persuade anyone the story’s elements “were the ingredients for a sure fire comedy smash”.
He added: “We do incredible things when we stand together – unite.”
In a special tribute to the late Lord Attenborough, who died in August, Prince William said in a recorded message: “His legacy is his inspiration to young film-makers.”
Actor Robert Downey Jr said: “As a director and mentor his passion was ceaseless.”
The obituary section also paid tribute to Robin Williams, Billie Whitelaw and Lauren Bacall – among others – who died in the past year.
Many viewers took to social media to voice surprise that Bob Hoskins was not included, however Bafta said he was remembered at its TV awards and in the film ceremony’s souvenir brochure.
“Bafta features individuals in televised obituaries only once, sadly due to the number of people we’d like to recognise at any one time, and that means difficult decisions have to be made as to which ceremony they should be included in,” Bafta told the BBC.
“As Bob died in April last year, just before the television awards, we felt it was right to remember his wonderful career then, rather than wait until last night’s film awards.”
The award for best animated film went to The Lego Movie, which caused a shock by being snubbed by the Oscars.
Its makers Phil Lord and Christopher Miller called the award “awesome” and joked: “You are our favourite academy by far.”
As previously announced, Mike Leigh was honoured with the Bafta Fellowship and BBC Films was honoured for outstanding contribution to British cinema.
Unbroken actor Jack O’Connell won the Rising Star award, the only one of the awards to be voted for by the public.
Northern Irish film Boogaloo and Graham won the short film award while the short animation prize went to The Bigger Picture.
Both shorts are nominated at the Oscars in two weeks.