The star, who was the son of Broadway legend Mary Martin (South Pacific, Peter Pan) and Texas attorney Benjamin Hagman, was 81. He is survived by his wife of nearly 58 years, Maj Hagman, their two children and their grandchildren.
"When he passed, he was surrounded by loved ones," his family said in a statement to the Dallas Morning News. "It was a peaceful passing, just as he had wished for. The family requests privacy at this time."
"This is so sad. Larry was really someone who was loved by everyone," his agent Joel Dean tells PEOPLE. "Me especially. He was the most loving, wonderful, generous man. And he was a true trouper."
In late 2011, Hagman, who had undergone a liver transplant in 1995, announced he had stage 2 throat cancer but had also signed on to star in the TNT reboot of Dallas. The show, which originally aired on CBS from 1978 to 1991, recently started filming its second season for the cable channel.
"Larry was back in his beloved Dallas, re-enacting the iconic role he loved most," said the family statement. According to the Morning News, Hagman's Dallas costars Linda Gray and Patrick Duffy were by his side at Medical City Dallas Hospital when he died.
In addition to portraying J.R. – a lovable, scheming, villainous oilman, whose shooting was a topic of international water-cooler discussion in 1980 – Hagman costarred with Barbara Eden as the astronaut Major Anthony Nelson on NBC's popular I Dream of Jeannie, which ran for five seasons starting in 1965.
"My deepest condolences go out to his wife Maj, his son and daughter and his grandchildren, as well as his friends in this time of his passing," Eden posted on her Facebook page. "I can honestly say that we've lost not just a great actor, not just a television icon, but an element of pure Americana."
Added Eden: "Goodbye Larry, there was no one like you before and there will never be anyone like you again."
Additional reporting by STEPHEN M. SILVERMAN