The trophies for Sir Sam Mendes's movie included best film, best British film, best director and best cinematography.
Joker won three awards including best actor for Joaquin Phoenix, while Renee Zellweger was named best actress for her portrayal of Judy Garland.
South Korean film Parasite won two prizes - for original screenplay and film not in the English language.
There were no major upsets or surprise winners, with 1917 unarguably dominating the evening.
"I couldn't be more thrilled," director Sir Sam told BBC News backstage. "There's the personal delight in seeing a story very close to me and my family be developed and enlarged but the massive thing has been audiences going in large numbers.
"None of us knew if an audience would turn up, it wasn't certain at all. It's coincided with awards season and the fact this is still number one in the UK after four weeks, [the awards have] really alerted people to the fact the movie is on, it rarely happens like that."
Sir Sam is the first British winner of best director at the Baftas since Danny Boyle won for Slumdog Millionaire in 2009.