Backwards-biographical story The Curious Case of Benjamin Button was named best new musical at the awards, Britain's equivalent of Broadway's Tony Awards.
Conclave star Lithgow added the Olivier to an awards shelf that already includes multiple Tony, Emmy and Golden Globe trophies, for depicting the author of Charlie and the Chocolate Factory in Mark Rosenblatt's play, which confronts Dahl's anti-Semitic views.
"I think I'm going to faint," said an emotional Lithgow, 79 on Sunday.
The US actor said he wanted to assure Britons that the transatlantic "special relationship is still firmly intact".
"It's not always easy to welcome an American into your midst, and at this particular moment, it's probably a little more complicated than usual," he said.
Lesley Manville, whose resume includes a stint as Princess Margaret in The Crown, took the best actress prize for her performance as shocked royal spouse Jocasta in Oedipus.
Director Robert Icke's modern-day reimagining of the ancient Greek tragedy — which opens on Broadway later this year — was named best revival of a play.
Imelda Staunton who played the Queen in the two final seasons of The Crown, won the fifth Olivier of her career, best actress in a musical, for Hello, Dolly!
Best actor in a musical went to John Dagleish as the titular man who ages in reverse in Benjamin Button.
The musical is based on a story by F Scott Fitzgerald that also inspired a 2008 movie starring Brad Pitt.
The Oliviers were handed out in a ceremony at London's Royal Albert Hall hosted by Broadway, TV and runway star Billy Porter and British soul singer Beverley Knight.
Stars in the audience included recent Academy Award winner Adrien Brody – a best-actor Olivier nominee for death-row stage drama The Fear of 13 – and Cate Blanchett, recently seen on the London stage in The Seagull.
Giant won three prizes including best new play.
Benjamin Button also won three, as did a boisterous outdoor production of Fiddler on the Roof at Regent's Park Open Air Theatre, which was named best musical revival.
Maimuna Memon was named best supporting actress in a musical for Tolstoy-inspired Natasha, Pierre & the Great Comet of 1812.
Layton Williams took the equivalent supporting actor prize for campy Celine Dion celebration Titanique.
For plays, supporting performer prizes went to Elliot Levey for Giant and Romola Garai for The Years.
Norwegian director Eline Arbo was named best director for The Years, an adaptation of Nobel Literature laureate Annie Ernaux's autobiographical book.
The prizes, which recognise achievements in theatre, opera and dance, were founded in 1976 and named for British actor Laurence Olivier.
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