Rexford was born in Conewango Valley, New York, to Kermit and Edith Rexford. His father's ownership of a Chevrolet dealership in the area gave him an early connection to automobiles, and he made his racing debut at the age of 16 at a local track in Leon, New York. At 18 he stepped away from racing to serve in the United States Navy during World War II, stationed in Newport, Rhode Island. After his discharge he returned home and resumed racing at short tracks in Buffalo, Rochester, and across Western New York.
When NASCAR introduced its "Strictly Stock" series in 1949, Rexford made three starts, recording his best finish — a third place — at Heidelberg Raceway. That showing convinced him to commit to full-time competition when the series was renamed the Grand National Series in 1950.
The 1950 Grand National season comprised 19 races and Rexford competed in 17 of them, driving Oldsmobiles, Fords, and Mercuries owned by Julian Buesink of Jamestown, New York. Rexford and teammate Lloyd Moore became the first Northern-born drivers to run the full circuit. Rexford took his first and only career win in the fifth race of the year at Canfield Speedway.
He spent most of the season battling for the points lead against Curtis Turner and a then-rookie Fireball Roberts. A fourth championship contender, Lee Petty, had been stripped of 809 points earlier in the season for competing in non-NASCAR-sanctioned races, effectively removing him from title contention.
The title came down to the final race at Occoneechee Speedway. Rexford held a slim points lead over Roberts, but his engine failed early in the race, leaving Roberts needing only a top-five finish to take the championship. Roberts' aggressive pursuit of a race win led his engine to also expire with under 50 laps remaining, handing Rexford the title in dramatic and controversial fashion.
At 23 years old, Rexford became the youngest champion in what would grow into the NASCAR Cup Series — a record that still stands.
The 1950 title proved to be Rexford's only full-time season at the Grand National level. He returned to the Northeast and competed part-time in the series through 1953, running his final Grand National race at Rochester, New York, where he finished fifth in the No. 60 Chevrolet for Julian Buesink.
After 1953, Rexford joined the Midwest Association for Race Car (MERC), the predecessor of what became ARCA. NASCAR founder Bill France Sr. responded by banning Rexford from NASCAR competition and fining him $1,000. Rexford refused to pay the fine and continued racing until 1956, retiring at the age of 29. He subsequently moved to Arizona, where he ran a trucking company for 25 years before retiring to California.
Rexford's 1950 championship carries several unique distinctions within NASCAR history. He is the only Cup Series champion to have achieved the title with just one career victory — a feat later matched by Ned Jarrett in 1961, Benny Parsons in 1973, and Matt Kenseth in 2003. He was the only series champion omitted from NASCAR's 50 Greatest Drivers list in 1998. Most strikingly, he remains the only eligible Cup Series champion from the 1949–2011 era not inducted into the NASCAR Hall of Fame, a status that has drawn persistent discussion among historians of the sport.
He died on April 18, 1994, in California, after a long illness.