The actor was found unresponsive by staff in his room at the Ritz-Carlton Hotel in Orlando, Florida on Sunday.
Officers from the Orange County Sheriff’s Department arrived after a call from hotel staff and pronounced him dead at the scene.
The exact details surrounding his death remain unclear.
But in a short statement posted to Twitter, the Sheriff’s Department said that detectives had found no signs of foul play or drug use during initial enquiries.
“We are devastated to confirm that our beloved Bob passed away today,” his family said in a statement.
“He was everything to us and we want you to know how much he loved his fans, performing live and bringing people from all walks of life together with laughter.”
Saget, a veteran stand-up comedian, had just started a new comedy tour across the US, performing as recently as Saturday in the city of Jacksonville, Florida.
After the show he had expressed his joy at being back on stage in a post on his Instagram page, telling fans that he was “happily addicted” to his craft.
“I’m back in comedy like I was when I was 26. I guess I’m finding my new voice and loving every moment of it,” he said.
Although Saget was idolised by many comedians for his profane and often cutting act, he was best known in the US for his role in the sitcom, Full House, where he starred alongside the actors John Stamos and fellow comic Dave Coulier.
He won plaudits for his portrayal of Danny Tanner, a widowed father of three daughters, playing the role from 1987 to 1995.
He reprised it in 2016 when Netflix revived the programme for a further five seasons.
In more recent years he found fame among younger generations as the narrator of the hit show How I Met Your Mother and for his guest stints as a highly fictionalised version of himself in the TV series, Entourage.
John Stamos, who worked with Saget on Full House and Fuller House, said he was “broken” at the news of his friend’s death.
“I am broken. I am gutted. I am in complete and utter shock. I will never ever have another friend like him. I love you so much Bobby,” Stamos posted on Twitter.
Whoopi Goldberg also paid tribute to her friend, giving a special mention to his “huge heart and abject lunacy”.
Accompanying a photo of himself and Saget, actor and writer Penn Jillette wrote on Twitter: “Saget would have wanted something tweeted that was really funny and in very bad taste. I can’t do that. I’m just so sad. Bob was just great on every level and we will all miss him.”
The former Daily Show host Jon Stewart called Saget “just the funniest and nicest” in a tweet on Sunday.
And comedian Judd Apatow said: “Bob Saget was so kind and when you spent time with him he made you laugh hard. He loved to be funny and he was hysterical. He was also there for everyone. A beautiful soul.” (Courtesy BBC)