The record haul follows the musical’s return to the West End last year, 25 years after it premiered in London.
Former Doctor Who co-stars David Tennant and Billie Piper won best actor and best actress in a play. Tennant for Richard II and Piper for Great Britain.
The awards, focusing predominantly on London theatre, are voted for entirely by members of the public.
The new production of Miss Saigon won nine of its 12 nominations, including best actor and actress in a musical, best direction, best choreography and best musical revival.
The musical, set during the Vietnam War and based on the opera Madame Butterfly, opened in London at the Prince Edward Theatre last May.
“I’ve always known that the public knows best,” said the show’s producer, Sir Cameron Mackintosh, adding that he was was particularly thrilled “by the extraordinary achievement of our four leading Asian performers winning the top acting awards”.
American newcomer Eva Noblezada, and Jon Jon Briones – who starred in the original cast 1989 London cast – took the awards for their leading roles.
Eighteen-year-old Noblezada’s rivals included singer Beverley Knight and Made In Dagenham’s Gemma Arterton, while Briones beat stars including Wet Wet Wet singer Marti Pellow and Dirty Rotten Scoundrels’ Robert Lindsay to the title.
There were also gongs for the best supporting actor and actress in a musical.
Sherlock’s Mark Gatiss won best supporting actor in a play for his role as the wily Menenius in the Donmar Warehouse’s production of Coriolanus.
Thor star Tom Hiddleston was nominated for his portrayal of Coriolanus in the best actor category, but lost out to Tennant as another of Shakespeare’s leading men, Richard II.
However, Coriolanus was the most successful play of the night, winning a further two awards for best play revival and lighting design.
Piper’s win – for her role as a tabloid editor in Great Britain, Richard Bean’s new play about hacking – was fiercely contested, with a shortlist that included Gillian Anderson’s Blanche DuBois, Helen McCrory as Medea and Imelda Staunton.
Lee Hall’s adaptation of the 1998 film Shakespeare in Love – currently playing at London’s Noel Coward Theatre – won best new play, while Memphis the Musical was named best new musical.
Sheffield Theatres’ Oliver! won the best regional production for the second consecutive year.
And there was a bittersweet win for Twickenham Theatre’s Sweeney Todd – in the best off-West End production. The theatre closed just days after the production completed its run, and is still looking for a new home.