Shaw was born in Shelbyville, Indiana. At sixteen he moved to Indianapolis, where he found work in a succession of automotive-related jobs. He made his first Indianapolis 500 appearance in 1927 and spent the next decade working his way to the front of American oval racing.
His three Indianapolis victories established him as the pre-eminent American driver of his era. He was the first driver to win the race in consecutive years, taking the 1939 and 1940 editions back-to-back in the Maserati 8CTF — an Italian car that ran under the name the "Boyle Special." The 1939 and 1940 results were particularly dominant performances that cemented his reputation as the master of the Indianapolis Motor Speedway.
In the 1941 race, Shaw was again a contender until a catastrophic wheel failure caused a crash that injured him and ended his competitive career at Indianapolis. Investigation after the race revealed that a defective wheel had been fitted to his car.
Shaw's most lasting contribution to motorsport came not from behind the wheel but in the years immediately after World War II. During the war, the Indianapolis Motor Speedway had been closed, and without maintenance the facility fell into severe disrepair. When Firestone hired Shaw to test a synthetic rubber tire at the circuit, he was dismayed by what he found.
He learned from then-owner Eddie Rickenbacker — the World War I flying ace and founder of Eastern Air Lines — that plans were being made to demolish the track and convert the land into a housing subdivision. Shaw immediately began approaching major automotive manufacturers to find a buyer willing to preserve the facility as a racing circuit, but each indicated they would convert it into a private test track.
Shaw then made contact with Terre Haute businessman Tony Hulman, heir to the Hulman & Company wholesale grocery and baking goods business (makers of Clabber Girl baking powder). Hulman was a lifelong motorsport fan who listened carefully to Shaw's case. Despite the visible decay when Shaw took him to see the property, Hulman purchased the speedway from Rickenbacker in November 1945 for $750,000.
As a reward for orchestrating the sale, Shaw was appointed president of the speedway, with full day-to-day operational authority. He brought to the role an expert's knowledge of the racing business that Hulman readily acknowledged he himself lacked. While Hulman invested heavily in physical improvements to the facility, Shaw rebuilt the race's reputation and public stature. The Indianapolis 500 of the late 1940s and early 1950s drew growing crowds each year, and the event's standing as the "Greatest Spectacle in Racing" was substantially rebuilt under the partnership of the two men.
Shaw also contributed to automotive journalism, serving as automotive test evaluator for Popular Science magazine, where his reports were noted for their accuracy and technical rigor compared to contemporaries writing for competing publications.
Shaw died in an airplane crash near Decatur, Indiana, on October 30, 1954 — one day before his fifty-second birthday. The pilot and one other passenger were also killed. He was survived by his wife, Cathleen "Boots" Stearns Shaw, and a nine-year-old son.
His autobiography, Gentlemen, Start Your Engines, was published in 1955 and covers his career through 1953.
Shaw was inducted into the Auto Racing Hall of Fame in 1963, the Automotive Hall of Fame in 1987, the National Sprint Car Hall of Fame in 1990, and both the International Motorsports Hall of Fame and Motorsports Hall of Fame of America in 1991. As of 2025, he remains the last Indiana native to win the Indianapolis 500.