The star won for best featured performance as an actress for playing the object of her uncle's lust in Arthur Miller's A View From a Bridge.
"Ever since I was a little girl I wanted to be on Broadway and here I am," Johansson, more widely known for such films as Match Point and Lost in Translation, said while accepting the award during a ceremony at New York's Radio City Music Hall.
Two socially conscious but radically different stories set in mid-20th century America, the exuberant rhythm'n'blues musical Memphis and anguished two-man drama Red, were big winners at the awards, which recognise achievement in live American theatre.
Red dominated, winning best director of a play for Michael Grandage and featured performance by an actor for Eddie Redmayne, as well as best lighting design for a play, best sound design and best scenic design.
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Best direction for a musical went to first-time nominee Terry Johnson of La Cage Aux Folles.
Memphis, a tale of segregation and integration in the American South, was cited for best orchestration, original score and best book of a musical, twice beating out Fela!, which won for best costume design of a musical and best sound design of a musical.
The ceremony, attended by Glee cast members Lea Michele and Matthew Morrison, Katie Holmes, Daniel Radcliffe, Will Smith and Jada Pinkett Smith, was hosted by Sean Hayes.
Hayes was up for leading actor in a musical for his portrayal of Chuck Baxter, the insecure company man who lends out his bachelor apartment for extra-marital, romantic dalliances by its executives, in a revival of Promises, Promises.
He was beaten by Levi Kreis as rock'n'roll wild man Jerry Lee Lewis in Million Dollar Quartet, but Hayes's co-star Katie Finneran won for best featured actress in a musical. Historically, Tony-nominated hosts have fared well. Hugh Jackman won a trophy in 2004 for his portrayal of Peter Allen in The Boy From Oz. Australian actor Anthony LaPaglia features in this year's Tony award telecast in the US, during a special presentation of Tony-nominated plays and play revivals. LaPaglia and co-star Tony Shalhoub will perform for Lend Me a Tenor.