Venturi, who had been hospitalised the last two months for a spinal infection, pneumonia and an intestinal infection, passed away on Friday afternoon in California.
He won the 1964 US Open at Congressional despite playing with severe dehydration and went on to become a respected television commentator with CBS Sports.
Venturi was unable to attend his induction to the Hall of Fame on May 6, with his sons Matt and Tim accepting on his behalf.
"I was very upset and saddened to hear the news of Ken's passing," 18-time major winner Nicklaus wrote on his Facebook page. "We all knew what a wonderful player Ken Venturi was, and how he fashioned a second successful career as an announcer.
"But far more important than how good he was at playing the game or covering it, Ken was my friend.
"Ken was fortunate in that the game of golf gave him so much, but without question Ken gave back far more to the game he loved than he ever gained from it. Over the years, Ken developed a circle of friends that is enormous and whose collective heart is heavy. All those in and out of the golf community will miss him, just as Barbara and I will.
"If there is some sense of fairness, it is that Ken was inducted into a Hall of Fame that he very much deserved to be in and, in fact, should have been in for many years.
"While I know he was not able to be there in person for his induction, I am certain there was an overwhelming sense of pride and peace that embraced Ken. It was a dream of Ken Venturi's that became a reality before he sadly left us."
A statement from Jack Peter, chief operating officer of the World Golf Hall of Fame read: "On behalf of the members, staff and volunteers of the World Golf Hall of Fame & Museum, we are saddened to learn of the passing of Ken Venturi. He was one of golf's iconic figures, and our thoughts and prayers go out to Ken's family.
"Ken made an unforgettable imprint on the game we love. He was a fantastic player and captivated the nation with his thrilling victory in the 1964 US Open. For 35 years in the broadcast booth at CBS, he was the warm, friendly voice millions invited into their homes to share his unique insights.
"To honour him, the United States flag at the Hall of Fame will be lowered to half-mast and a special tribute will be created in the Museum. When Ken learned he would be a part of the Class of 2013, he said 'The greatest reward in life is to be remembered.'
"The Hall of Fame and golf fans everywhere will never forget the impact Ken had on the game."