Bignotti was born in San Mateo, California. He established himself as a successful mechanic in California midget car racing in the late 1940s, winning championships as chief mechanic for driver Fred Agabashian and enabling himself to leave a job as a shipbuilder. When the popularity of midget-car racing declined at the end of the decade, Bignotti stepped away from the sport for a period and worked in his mother-in-law's flower shop. He returned to the chief mechanic role in 1954 when Agabashian recruited him for that year's Indianapolis 500; Agabashian finished sixth, and Bignotti became a near-permanent fixture in American open-wheel racing.
In 1956 Bignotti joined forces with Bob Bowes, president of the Bowes Seal-Fast Corporation, forming Bignotti-Bowes Racing. The team had moderate success with drivers Johnny Boyd and Jud Larson in its early years. The partnership's fortunes changed dramatically in 1960 when A. J. Foyt joined as driver. Bignotti and Foyt proved a dominant combination, winning four USAC championships and 27 races together. The 1964 season was their high watermark: the team won ten of the thirteen championship races. Despite their extraordinary on-track success, the two men clashed frequently, and Bignotti departed his role several times before the partnership permanently dissolved mid-season in 1965.
For the 1966 Indianapolis 500 Bignotti served as chief mechanic for the team operated by oil magnate John W. Mecom Jr. Graham Hill won the race for the team; teammate Jackie Stewart led the most laps before a mechanical failure ended his day.
Bignotti then joined Vel's Parnelli Jones Racing, where his partnership with Al Unser produced some of his most celebrated results. He won the 1970 and 1971 Indianapolis 500 with Unser, along with the 1970 USAC Championship — the team won ten races that season, equaling the record Bignotti had set with Foyt in 1964. Further USAC championships followed with Joe Leonard in 1971 and 1972.
Beginning in 1973, Bignotti joined Patrick Racing, where Gordon Johncock won the 1973 Indianapolis 500 under his preparation. Bignotti also guided Patrick to the 1976 USAC Championship.
In the early 1980s Bignotti co-founded Bignotti-Cotter Racing with True Value CEO Dan Cotter. With Tom Sneva driving, the team won the 1983 Indianapolis 500 — Bignotti's seventh and final win at the Speedway. The team then ran Roberto Guerrero in the CART championship for three further seasons before folding, at which point Bignotti retired from racing. His older brother Al had worked alongside him as a mechanic at the 1966, 1970, 1971, and 1983 Indianapolis 500 victories.
In retirement Bignotti maintained friendships with many former drivers, including A. J. Foyt, with whom he had reconciled long after the conflicts of their working relationship. He was an avid golfer and played until the age of 90. Bignotti died in his sleep at his home in Las Vegas, Nevada, on September 27, 2013, at the age of 97, and was buried at Crown Hill Cemetery in Indianapolis, Indiana.
Bignotti's record of seven Indianapolis 500 victories as chief mechanic places him among the most accomplished figures in the race's history. He was inducted into multiple halls of fame: the Indianapolis Motor Speedway Hall of Fame (1975), the International Motorsports Hall of Fame (1993), the Motorsports Hall of Fame of America (1993), the National Midget Auto Racing Hall of Fame (2003), the USAC Hall of Fame (2014), and the West Coast Stock Car/Motorsports Hall of Fame (2018). His nine USAC championship titles across partnerships with Foyt, Unser, Leonard, Johncock, and Sneva reflect a breadth of talent that extended well beyond any single driver relationship.