Flaherty began his racing career in 1946, competing in track roadsters on the regional circuit. In 1948 he relocated to the Midwest to join Andy Granatelli's Hurricane Hot Rod Association, the organisation through which Granatelli developed drivers and built a presence in American open-wheel competition. That association gave Flaherty access to competitive machinery and the oval racing environment that would define his career.
Flaherty competed in the national championship car series — first under AAA sanction, then USAC — across the 1950, 1953–1956, 1958–1959, and 1963 seasons. Over 19 championship car starts he finished in the top ten nine times, demonstrating consistent competitiveness against the leading drivers of American oval racing.
His victories at Milwaukee in 1955 and 1956 established him as a legitimate front-runner ahead of his breakthrough at Indianapolis. The 1955 Milwaukee win placed him eighth in the season standings with 790 points; the 1956 campaign, anchored by his Indianapolis victory and another Milwaukee win, saw him finish second in the USAC championship with 1,500 points.
Flaherty's Indianapolis 500 appearances spanned five races in 1950, 1953, 1955, 1956, and 1959. He also entered in 1954 and 1958 but did not qualify for those events. In 1963, at the age of 37, he made a final championship car entry at Milwaukee.
Flaherty's 1956 Indianapolis 500 victory was the centrepiece of his career. He started from pole position and led 127 of the 200 laps, controlling the race with a level of dominance that placed the result among the most commanding Indianapolis wins of the decade. He drove a Watson roadster, the type of machinery built by mechanic A.J. Watson that became the dominant chassis at Indianapolis in the late 1950s.
The victory earned Flaherty eight World Drivers' Championship points, as the Indianapolis 500 was included in the FIA World Drivers' Championship from 1950 through 1960. His five Indy starts during that period — with one pole position and one win — constitute his entire World Championship record. He accumulated eight points from those five eligible appearances.
Less than three months after winning at Indianapolis in May 1956, Flaherty was severely injured in a racing accident. The extent of his injuries prevented him from competing at the 1957 Indianapolis 500, interrupting what had been the most productive period of his career at the worst possible moment. He returned to championship car racing in 1958 and 1959, but the injury effectively ended his prospects of adding another major title.
Flaherty made one appearance in NASCAR, contesting a single race in 1951. The outing placed him outside his primary discipline but reflected the crossover participation that was common among American oval specialists of the era.
After retiring from racing, Flaherty turned to other competitive pursuits. He raced pigeons competitively for more than twenty years, becoming accomplished in a hobby that requires considerable skill in breeding, training, and race strategy. He also built and operated a portable go-kart track, taking on all-comers in informal competition and maintaining his connection to competitive driving in a low-key and accessible format. He died in Oxnard, California on April 9, 2002, at the age of 76.
Flaherty's 1956 Indianapolis 500 victory secured him a permanent place in the record of the race that serves as American oval racing's defining event. His induction into the Auto Racing Hall of Fame in 2006 recognised a career built on genuine competitiveness across the AAA and USAC eras, a Milwaukee double in two consecutive years, and the pole-to-victory run at Indianapolis that stands as his lasting contribution to the sport.