Green Book was the night's other big film winner, scooping three prizes.
It won best musical or comedy film, best screenplay and best supporting actor for Mahershala Ali.
Set in the 1960s and co-starring Viggo Mortensen, Green Book tells the story of an African-American musician who hires an Italian-American driver to take him on a tour of the Deep South.
Its Golden Globes victories mark it out as one of the favourites for next month's Oscars.
Bohemian Rhapsody's wins come despite mixed reviews and upheaval during production.
Its original director Bryan Singer was fired from the film due to "unreliable behaviour" amid reports that he had clashed with Malek on set. Dexter Fletcher was brought in to complete the project.
In his acceptance speech, Malek thanked Queen and said he was "beyond moved" to have won.
He dedicated the award to Mercury, saying: "This is for and because of you, gorgeous!"
One of the night's other big surprises was Glenn Close winning best drama actress for her role in The Wife.
She beat off competition from Lady Gaga, who had been widely expected to win for A Star Is Born.
In an impassioned speech, Close spoke about the importance of women following their dreams and not just looking after their home lives.
She said: "Women, we are nurturers, that is what is expected of us. We have our husband and our children and we have to find personal fulfilment and follow our dreams and we have to say, 'I can do that' and, 'I should be allowed to do that'."
The British winners included Olivia Colman, who won best actress (comedy or musical) for her portrayal of Queen Anne in The Favourite.
Christian Bale was also rewarded for his role as former vice president Dick Cheney - which scooped him best actor (comedy or musical).
British actors Richard Madden and Ben Whishaw won TV awards for their roles in BBC dramas Bodyguard and A Very English Scandal respectively.
In the other TV categories, The Americans was named best drama and The Kominsky Method picked up best TV comedy. The Netflix show's star Michael Douglas won best actor in a TV series (musical or comedy) for playing an ageing acting coach.
He dedicated his award to his "extraordinary, wonderful wife Catherine" - that's Catherine Zeta Jones - and his 102-year-old father Kirk.
The award came 49 years after he received his first Golden Globe nomination for best male newcomer.
Sandra Oh won best actress in a TV drama for Killing Eve, the unconventional crime drama that started out on BBC America. She thanked her Korean parents, who were seated in the audience, in an emotional acceptance speech.
Oh also co-hosted Sunday's ceremony in Beverly Hills alongside Brooklyn Nine-Nine actor Andy Samberg.
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