Dandurand was born in Bourbonnais, Illinois. He moved to Canada with his family when he was 16 years old and attended Collège Sainte-Marie de Montréal. He was a referee in the National Hockey Association and was involved with minor hockey in the Montreal, Quebec area.
On November 2, 1921, Dandurand and his partners, Joseph Cattarinich and Louis Letourneau, purchased the Montreal Canadiens hockey club from the widow of George Kennedy for $11,000. Under Dandurand's ownership, the Canadiens won the Stanley Cup in 1924, 1930, and 1931. Letourneau sold his stake in the club in 1930, and Dandurand and Cattarinich continued as owners until selling the team in 1935 for $165,000.
Along with hockey, Dandurand and his partners were heavily involved with horse racing. In 1932, they bought Blue Bonnets Raceway in Montreal. They owned 17 tracks in Quebec, Ontario, New York, Ohio, Delaware, Illinois, Utah, and Louisiana at the time of Cattarinich's death in 1938. Dandurand was also a boxing and wrestling promoter in Montreal and a director of the Montreal Royals baseball team.
In 1946, Dandurand founded the Montreal Alouettes football team with Eric Cradock and Lew Hayman. The team played in the Interprovincial Rugby Football Union, which later became the CFL's east division.
In his later years, Dandurand owned a successful restaurant in downtown Montreal. He was inducted into the Hockey Hall of Fame in 1963 and died of a heart attack on June 26, 1964 at age 75.
The Leo Dandurand Trophy is a CFL award presented each year to the most outstanding lineman in the East Division.