Redmayne thanked his “staggering partner in crime”, co-star Felicity Jones, and his “ferocious but incredibly kind director James Marsh”.
Dark comedy Birdman won best film and best director for Mexican film-maker Alejandro Gonzalez Inarritu.
It also won best cinematography and best original screenplay.
The film sees Michael Keaton play a former movie superhero actor, who hopes to revive his washed-up career by putting on a Broadway play.
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.
“I’m so happy, I’m thrilled that we were able to shine a light on Alzheimer’s disease,” she said.
“So many people who have this disease feel marginalised. People who have Alzheimer’s disease deserve to be seen so we can find a cure.”
Redmayne was honoured for his portrayal of physicist Stephen Hawking, who has motor neurone disease (ALS).
Accepting his award, he 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”.