It was the 80-year-old Lithgow's third Tony, having won his first 53 years ago for his Broadway debut in The Changing Room.
"At every point we had to figure out, 'Why is this man doing this?'" he told reporters.
"Anti-Semitism, cruelty of all kinds ... these are things that we're dealing with these days up front and personal ... that's what makes Giant so important and such a success."
Rose Byrne was hoping to pick up her first Tony for her starring role in a Broadway revival of a century-old Noel Coward play but lost to Lesley Manville in Oedipus.
The Australian was is nominated for best actress in a leading role in a play for her critically-acclaimed portrayal of Jane Banbury in a the 1925 comedy of manners, Fallen Angels.
This year's awards cap a record Broadway season, having generated $US1.91 billion ($A2.71 billion) in grosses as audiences turned out for both established hits and new productions.
Pop singer Pink, hosting the ceremony at Radio City Music Hall, led a huge opening number with a Broadway-updated rendition of Lady Marmalade and starring the cast of every single nominated musical.
Zhailon Levingston and Bill Rauch won best direction of a musical for Cats: The Jellicle Ball, a bold reimagining of Andrew Lloyd Webber's musical that places the story within New York's ballroom culture.
Joe Mantello won best direction of a play for the revival of Arthur Miller's Death of a Salesman, starring Nathan Lane and Laurie Metcalf.
Metcalf won best featured actress in a play for her role as Linda Loman in Death of a Salesman.
Metcalf said she had never seen a production of Salesman in anticipation of one day starring in it.
It was her third Tony.
New musicals The Lost Boys and Schmigadoon! led all nominees with 12 nods each, followed by the revival Ragtime with 11.
