Snyder, who was 69, died at his Phoenix home Monday, the university said. He coached the Sun Devils from 1992-2000. His 1996 team went 11-0 in the regular season before a last-minute 20-17 loss to Ohio State in the Rose Bowl.
Snyder also was the coach at Utah State from 1976-82 and California from 1987-91. His overall record was 126-105-5. At Arizona State, Snyder was 58-47, second only to Frank Kush for victories with the Sun Devils.
He was an assistant coach for the NFL's Los Angeles Rams from 1983-86.
"He was a guy that helped a lot of us kids grow up, especially myself," said Keith Poole, a wide receiver on that 1996 team. "It was my first time away from home and he basically took over as a father. He taught you how to be tough. He didn't have any soft love. You learned to respect him for that."
Snyder, who left coaching after he was fired by Arizona State in 2000, was diagnosed with cancer last June.
He told The Arizona Republic last fall that, "if all of a sudden I'm gone in a year, it's been a damn good life. And I've lived a healthier, longer life than most."
Snyder's 1996 team, which upset two-time defending national champion and then-No. 1 Nebraska 19-0, had Jake Plummer at quarterback and Pat Tillman at linebacker.
Plummer went on to quarterback the NFL's Arizona Cardinals and Denver Broncos. Tillman played safety for the NFL's Arizona Cardinals, then quit the sport to join the Army Rangers. He was killed by friendly fire in Afghanistan in 2004.
Other players on that squad who went on to play in the NFL included Poole, Juan Roque, Lenzie Jackson, Grey Ruegamer, Steve Bush, Derrick Rodgers, Derek Smith and Jason Simmons.
"He taught us how to care for each other," Poole said. "I think that's why we were so good that year."
Snyder won the Paul "Bear" Bryant Award as national coach of the year for the 1996 season.
His 1997 team was 9-3 and beat Iowa in the Sun Bowl, the Sun Devils' first bowl victory in a decade.
Snyder was Pac-10 coach of the year for California in 1990 and Arizona State in 1996.
Seven of his Arizona State players were first-round draft picks -- Shante Carver, Craig Newsome, Erik Flowers, Adam Archuleta, Todd Heap, Levi Jones and Terrell Suggs.
Snyder was 39-37-1 at Utah State, coaching the Aggies to two conference championships. At Cal, he was 29-24-4., His 1990 Cal team went 7-4-1 and appeared in a bowl game -- the school's second since 1958. In 1991, he led the Bears to a 10-2 record and a victory over ACC champion Clemson in the Citrus Bowl.
Snyder is survived by his wife, three daughters, a son-in-law, two grandchildren, four sisters, two brothers and -- in his family's words -- his "beloved" dog Ella.
A private memorial service is planned.