The Williams F1 team, founded in 1977, dominated much of the 1980s and 1990s under his guidance, and has 117 race victories in total.
The team built in his name grew into one of the most successful in the sport, winning seven drivers' titles and nine constructors' championships.
"It is with great sadness that on behalf of the Williams family, the team can confirm the death of Sir Frank Williams CBE, Founder and Former Team Principal of Williams Racing, at the age of 79," the team said.
"After being admitted into hospital on Friday, Sir Frank passed away peacefully this morning surrounded by his family.
"Today we pay tribute to our much loved and inspirational figurehead. Frank will be sorely missed. We request that all friends and colleagues respect the Williams family's wishes for privacy at this time."
Williams sustained a spinal cord injury in a 1986 car crash which left him unable to walk.
He stepped down from the Williams' board in 2012, and handed over day-to-day running of the team to his daughter Claire in 2013.
Williams and his family sold the team to investment firm Dorilton Capital in the last 12 months.
Current Williams F1 drivers George Russell and Nicholas Latifi paid their respects to Williams on Twitter.
"Today, we say goodbye to the man who defined our team," Russell said. "Sir Frank was such a genuinely wonderful human being and I'll always remember the laughs we shared. He was more than a boss, he was a mentor and a friend to everybody who joined the Williams Racing family and so many others.
"It has been a genuine honour racing for him and being a small part of the incredible legacy he leaves behind, a legacy that will forever live on in the heart and soul of this team. Rest in peace, Sir Frank. Thank you for everything."
Latifi said: "RIP Sir Frank Williams. Such sad news. A huge loss for our sport and our team.
"It's been an honour to represent your name on the world stage and we will continue to push hard to take the team back up the grid."
Williams' current CEO and team principal Jost Capito paid tribute to the team's founder.
"The Williams Racing team is truly saddened by the passing of our founder Sir Frank Williams," he said.
"Sir Frank was a legend and icon of our sport. His passing marks the end of an era for our team and for the sport of Formula 1. He was one of a kind and a true pioneer. Despite considerable adversity in his life, he led our team to 16 World Championships, making us one of the most successful teams in the history of the sport.
"His values including integrity, teamwork and a fierce independence and determination, remain the core ethos of our team and are his legacy, as is the Williams family name under which we proudly race. Our thoughts are with the Williams family at this difficult time."
F1 President and CEO Stefano Domenicali also paid his respects.
"He was a true giant of our sport that overcame the most difficult of challenges in life and battled every day to win on and off the track. We have lost a much-loved and respected member of the F1 family and he will be hugely missed," Domenicali said.
"His incredible achievements and personality will be etched on our sport forever. My thoughts are with all the Williams family and friends at this sad time."