It also won best cinematography and best original screenplay.
The movie sees Michael Keaton play a former movie superhero actor, who hopes to revive his washed-up career by putting on a Broadway play.
Eddie Redmayne has won the best actor Oscar for The Theory of Everything, while Julianne Moore picked up best actress for Still Alice.
The British actor thanked his “staggering partner in crime”, co-star Felicity Jones, and his “ferocious but incredibly kind director James Marsh”.
Julianne Moore used her speech to raise awareness for Alzheimer’s disease – in Still Alice, she plays a 50-year-old who has early on-set Alzheimer’s.
Eddie Redmayne was honored for his portrayal of physicist Stephen Hawking, who has motor neurone disease (ALS).
Accepting his award, the actor thanked the Hawking family, including Jane Hawking on whose book the film is based, and said his award belonged “to all of the people around the world battling ALS”.
Richard Linklater’s Boyhood won just one award from six nominations – best supporting actress – which went to Patricia Arquette.
Patricia Arquette thanked “her Boyhood family” and “every woman who gave birth”.
“To every woman… we have fought for everybody else’s equal rights. It’s our time to have wage equality,” she added, to huge applause from the audience.
JK Simmons won best supporting actor for Whiplash, in which he played a strict drumming teacher at a music conservatory.
Whiplash also won the award for best editing and best sound mixing.
The Grand Budapest Hotel picked up best costume design (Milena Canonero), as well as best hair and make-up (British duo Frances Hannon and Mark Coulier). It also won best score and production design.
Frances Hannon thanked absent actor Bill Murray – who has a cameo in the film – for introducing her to director Wes Anderson on the set of his film Rushmore 17 years earlier.
Matt Kirkby and James Lucas, picked up the award for best live short action film, The Phone Call, starring Sally Hawkins and Jim Broadbent.
Alan Turing drama The Imitation Game, starring Benedict Cumberbatch, won best adapted screenplay.
Best foreign language film went to Polish black and white family drama Ida.
John Legend and Common’s track Glory, from civil rights drama Selma, won best song.
Emmanuel Lubezko’s win for best cinematography for Birdman was his second Oscar in as many years – in 2014, he won the same award for his work on Gravity.
Citizenfour, which chronicles one of the biggest intelligence leaks in American history, won best documentary feature.
It shows former NSA contractor Edward Snowden at the very moment he made his sensational revelations detailing extensive internet and phone surveillance by the US government.
Best documentary short was won by Crisis Hotline: Veterans Press 1, about the counselors who work with military veterans on a 24-hour phone helpline.
Clint Eastwood’s Iraq war drama American Sniper won the award for best sound editing.
The 87th Academy Awards took place at Hollywood’s 3,300-seat Dolby Theatre.
Performers at this year’s ceremony included Lady Gaga – who sang a medley of Sound of Music songs to celebrate the classic film’s 50th year – Jennifer Hudson and Anna Kendrick.
Host Neil Patrick Harris kicked off with a song which paid homage to Hollywood’s film industry, accompanied by Kendrick and Jack Black.
There were eight contenders for best picture:
Birdman and Wes Anderson’s quirky comedy The Grand Budapest Hotel began the night with nine nominations each. The Imitation Game had eight.
Clint Eastwood’s American Sniper and Boyhood had six apiece.
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