Sneva's first powered vehicle was a go-kart received at age fourteen. By eighteen he was racing stock cars. He graduated from Eastern Washington State College with an education degree, taught mathematics and physical education, drove a school bus, and coached football and basketball — before motorsport eventually won out. He was the eldest of five racing brothers; the next oldest, Jerry, also competed at Indianapolis. Another brother, Babe, died in 1976 from head injuries sustained in a racing accident in British Columbia.
Sneva made his transition to Indy car racing through the USAC ranks, developing a reputation as a brilliant qualifier with a naturally aggressive style suited to the bias-ply tires of the era.
Sneva's rise to prominence came driving for Roger Penske. On May 14, 1977, at the Indianapolis Motor Speedway, he became the first driver in history to qualify for the Indianapolis 500 at 200 mph or faster, setting a one-lap record of 200.535 mph in his Norton Spirit McLaren M24/Cosworth. That season he won two races, including a Penske 1-2 finish at the Pocono 500, and claimed the USAC National Championship — the first in Team Penske's history.
In 1978, despite not winning a race, Sneva's consistency — five second-place finishes and sixteen top-fives — secured him a second consecutive championship. Penske released him after the season, a parting Penske later attributed to disagreements over team direction rather than results.
Sneva's relationship with the Indianapolis 500 was defined for years by spectacular near-misses. In 1975, during his second Indy 500, he was involved in one of the race's most famous crashes: after touching wheels with Eldon Rasmussen, his car went into the catch fence and tore apart, leaving him with burns on fifteen percent of his body. He walked to the ambulance but spent time in intensive care primarily for lung issues from fire retardant. He later quipped about his instinct in the moment: "In a situation like that it's important to talk to yourself: 'Faint, you coward, faint!'"
He finished second at Indianapolis in 1977, 1978, and 1980. His 1980 second-place result is notable as one of only two occasions in the race's history when a driver starting last (33rd position) finished as runner-up and one of only three times a last-starting driver led laps during the race. In 1981, charging from 20th on the grid, he led early before mechanical failure intervened. In 1982, he was battling eventual winner Gordon Johncock and Rick Mears for victory when his engine began losing power near the finish.
In 1984, driving the Texaco Star March 84C/Cosworth for Mayer Motor Racing, he became the first driver to qualify at Indianapolis over 210 mph, posting a one-lap record of 210.689 mph.
The breakthrough came in 1983, driving the Texaco Star March 83C/Cosworth for Bignotti-Cotter Racing. Sneva engaged in a thrilling late-race duel with Penske driver Al Unser, with Al Unser Jr. — Unser's rookie son — running as a lapped car in the mix. Sneva passed both and won, in what also proved to be the last of legendary chief mechanic George Bignotti's record seven Indianapolis 500 victories. The win gave Sneva his nickname: "the Gas Man." For Sneva, the victory carried the particular sweetness of triumph over the team that had dismissed him five years earlier.
As defending champion in 1984, he dueled with Rick Mears inside the final 32 laps before a CV joint failure ended his bid for a second title. In 1985, he drove an ill-handling car to a second-place finish before being taken out in a crash with a lapped car.
Sneva's results declined in the second half of the 1980s, a decline some observers linked to the series' transition to radial tires between 1985 and 1987 — a tire type less compatible with his aggressive driving style. He retired after his final start at the 1992 Indianapolis 500, finishing his career with thirteen Indy car victories and fourteen pole positions.
After retiring from driving, Sneva worked as a color commentator for ABC's Wide World of Sports, calling several Indianapolis 500s. He later became involved in the golf course business in Paradise Valley, Arizona. He was inducted into the Indianapolis Motor Speedway Hall of Fame in 2004 and the Motorsports Hall of Fame of America in 2005.