He suffered a heart attack in Daytona Beach in Florida where he was due to play a concert on Wednesday evening.
He followed his father in the music business as a teenager, and later became his father's musical director and conductor.
He was famously kidnapped and held for ransom when he was 19.
A gang seized him at gunpoint at a casino in Lake Tahoe. His father paid a $240,000 (Â£168,000) ransom, which was later recovered after the kidnappers were arrested.
When he died on Wednesday, Frank Junior was due to perform at the Peabody Auditorium as part of his Sinatra Sings Sinatra tour, which included some of his father's greatest hits interwoven with family reminiscences.
The venue announced on its Facebook page: "Frank Sinatra Jr has passed away. Our love to his family and friends."
His sister Nancy, who also followed their father into the music business, said on her Facebook page: "Sleep warm, Frankie."
Tony Bennett wrote on Twitter: "I am deeply saddened to hear of the passing of Frank Sinatra Jr and send my sympathy to Tina and Nancy and the Sinatra family."
TV and radio host Larry King tweeted: "I always admired Frank Sinatra Jr. He was born with a hard name to live up to, but I loved his talent and his ability to be forthright."
Frank Sinatra Jr released six full-length albums and often made television appearances, recently starring as himself in two episodes of Family Guy.
Seth MacFarlane, who created the animated comedy, wrote: "Frank Sinatra, Jr. was a friend to Family Guy, and a friend to me. I'm saddened at his passing, but grateful to have known him."
Mia Farrow said: "RIP Frank Sinatra Jr. Thoughts with his mother Nancy, sisters Nancy and Tina and his son Michael."
Frank Sinatra Jr began singing with the Tommy Dorsey band, a version of the ensemble the elder Sinatra had sung with more than two decades before, at the age of 19.
The New York Times wrote in 1963: "Young Mr Sinatra has taken careful note of every aspect of his father's singing.
"He knows - and projects - the inflections, the shading, the phrasing that his father used."
Last year, he performed the national anthem at Los Angeles Dodgers and New York Yankees games in celebration of the centennial of his father's birth.
Frank Senior died of a heart attack in 1998.
The American singer, Frank Sinatra Junior, who carried on his father's legacy with his own career in music, has died at the age of 72.