The four child actresses who alternately played the titular heroine shared the Best Actress in a Musical trophy, while Bertie Carvel, who plays the headmistress of the girl's school, was named Best Actor in a Musical.
Matthew Warchus was declared Best Director for helming the show, which also earned prizes for Best Set Design, Best Theatre Choreographer and Best Sound Design.
"Crazy For You" won the awards for Best Musical Revival and Best Costume Design.
Eugene O'Neill's "Anna Christie" won the Best Revival of a Play honor, as well as the Best Actress gong for star Ruth Wilson. Benedict Cumberbatch and Jonny Lee Miller shared the Best Actor in a Play award for their roles-swapping performances in "Frankenstein."
The Best New Play prize went to John Hodge's black comedy "Collaborators."