The British comedian’s victory was one of the biggest surprises of the evening as he beat a host of Hollywood heavyweights to the best leading actor in a drama prize.
The 33-year-old could not hide his shock as he received the award for his performance in the hit National Theatre show One Man, Two Guvnors.
Fighting back tears, Corden paid an emotional tribute to his girlfriend Julia, telling the audience he was "overwhelmed" at the honour.
He said it was a great honour to be mentioned in the same breath as actors such as James Earl Jones, Frank Langella, John Lithgow and Philip Seymour Hoffman.
Corden said: "To be on a list with you was enough and holding this, it honestly just reminds me that there is no such thing as best.
"Honestly, I am overwhelmed."
Corden began to well up when he thanked his girlfriend, who was in the audience apologising to the guests and joking, "I'm so sorry, this is ridiculous. It's good that it's not hot in here, isn't it."
He told them: "My girlfriend Julia gave birth to our son five days before we started rehearsals. She is my baby mama and I can't wait to marry her.
"Seriously, I would not be holding this if it wasn't for her. She made me say 'us' instead of 'I' and 'we' instead of 'me', and I love her. Thank you very much."
He said he shared the award with the show's "brilliant" cast and crew, and thanked producer Bob Boyett, "who took a huge risk bringing this ridiculously silly play to your city".
He told the appreciative crowd: "Your city, the audiences here who have embraced our show like we never ever imagined they could, every night is a joy."
The tearful performance means Corden joins the likes of Kate Winslet and Gwyneth Paltrow, both of whom were ridiculed for their Golden Globe and Oscar acceptance speeches in 2009 and 1999 respectively.
The 66th Tony Awards were held at the Beacon Theatre in New York to celebrate excellence in Broadway theatre.
It was a big event for stars from this side of the Atlantic, with a stage version of the low-budget Irish film hit Once winning nine out of 12 nominations.
But while Corden celebrated with another award to add to his mantelpiece, British stars such as Andrew Lloyd Webber and Sir Tim Rice missed out.
British director John Tiffany picked up the award today for best direction of a musical for bittersweet production Once, his Broadway debut.
Playwright Enda Walsh also took home the award for best book of a musical for the same show, while fellow Irishman Bob Crowley was recognised for scenic design and Martin Lowe for best orchestration.
Despite having two nominations – for Jesus Christ Superstar and Evita – Lloyd Webber and Sir Tim were beaten to the gong by The Gershwins' Porgy And Bess.
Former Coronation Street actress Tracie Bennett missed out on an award for her role as Judy Garland in End Of The Rainbow.
