The Gavin & Stacy star held back tears as he picked up the best leading actor in a drama gong for his hit theatre show One Man, Two Guvnors at the Big Apple ceremony.
James, who won acclaim for the London run of the show, welled up as he thanked his other half.
He said: 'My girlfriend Julia gave birth to our son like, five days before we started rehearsals.
"She's my baby mama, and I can't wait to marry her. Seriously, I would not be holding this if it wasn't for her."
He tearfully added: "She makes me say 'us' instead of 'I' and 'we' instead of 'me', and I love her. Thank you very much."
The Brit beat a host of Hollywood legends to the trophy.
Big names he was up against included James Earl Jones, Frank Langella, John Lithgow and Philip Seymour Hoffman.
He said he was "overwhelmed" at being nominated alongside such highly-regarded actors.
"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," he said.
The Tony Awards are an annual celebration of excellence of Broadway theatre.
Fellow winners at the cermony included Hugh Jackman, who was surprised when he his wife Deborra-Lee Furness emerged to present him with a special award one-off award recognising his contribution to Broadway and his charity work.
"She's never kept a secret her entire life," said the actor as he took to the stage. "[She said] 'I'm off to the loo,' and I was like, 'OK, see you in a bit.'"