Hagman, who had suffered from cancer and liver disease, died in hospital in Dallas, Texas, the Dallas Morning News reported.
Linda Gray, a long-time friend who played Hagman's on-screen wife, Sue Ellen Ewing, was by his bedside when he died.
His family said in a statement carried by the Morning News: "Larry was back in his beloved Dallas re-enacting the iconic role he loved most
"Larry's family and close friends had joined him in Dallas for the Thanksgiving holiday. When he passed, he was surrounded by loved ones. It was a peaceful passing, just as he had wished for. The family requests privacy at this time."
In a statement from Gray's agent she said: "Larry Hagman was my best friend for 35 years.
"He was the Pied Piper of life and brought joy to everyone he knew. He was creative, generous, funny, loving and talented and I will miss him enormously. He was an original and lived life to the full."
Tributes from around the world flooded on to Twitter.
Hagman's Dallas co-star Ken Kercheval, who played Cliff Barnes, tweeted: "A friend and long time partner... the other half...RIP Larry Hagman.... your spirit will live long."
Actor William Shatner also paid tribute with the words: "My thoughts and prayers go out to the family of Larry Hagman. My best, Bill."
Veteran American broadcaster Larry King wrote: "I'm shocked. Larry Hagman was a dear man who had an incredible career. He helped me to stop smoking. He really was a very special person."
Journalist and broadcaster Piers Morgan tweeted: "Very sad to hear that Larry Hagman has died. His JR Ewing character was the greatest TV villain of them all. Wonderful actor."
Actress Elizabeth Hurley also wrote: "RIP Larry Hagman. Thank you for being such an entertaining actor and giving us such happy memories."
Meanwhile, Barbara Eden, who starred with Hagman in 1960s' sitcom I Dream of Jeannie, published a statement on her Facebook page which read: "I can honestly say that we've lost not just a great actor, not just a television icon, but an element of pure Americana.
"Goodbye Larry, there was no one like you before and there will never be anyone like you again."
Hagman was born in Fort Worth, Texas on September 21 1931, the son of actress Mary Martin and lawyer Ben Hagman, a biography on his official website said.
While in England with the US Air Force he met and married his wife of almost 60 years, Swedish designer Maj Axelsson. The couple later had two children.
He became a star in 1965 in the TV comedy series I Dream of Jeannie, in which he played an astronaut haunted by the beautiful blonde genie, played by Barbara Eden.
But it was in 1977 when he landed the role of merciless oil magnate JR Ewing, the character at the centre of the show Dallas, that his worldwide fame was cemented.
The series ran for 13 seasons and on November 21, 1980 more than 350 million people tuned in to find out "who shot JR".
Hagman refused to be defined by his most enduring role, acting in films such as Nixon and Primary Colors.
But he also had health problems. In 1992 he was diagnosed with cirrhosis of the liver and three years later he had a liver transplant.
In October last year he discovered a tumour on his tongue and was diagnosed with cancer, and underwent six weeks of chemotherapy and radiation before it went into remission in March.
Earlier this year he appeared in a new 10-episode series of Dallas, with a second series in production and due to run next year.