Born in Saint-Barnabé-Sud, Quebec, Côté was training to be a boxer when he switched to running marathons. He competed in his first Boston Marathon in 1936 and won the race in 1940, 1943, 1944, and 1948. He set a new course record with his 1940 victory, and was awarded the Lou Marsh Trophy as Canada's top athlete of the year. Côté was the first francophone recipient of the award.