In 1974, when he was 31, Cordero won the Kentucky Derby aboard Cannonade. He won the Derby twice more, making him one of only eight jockeys to win the race three or more times in the Derby's history. In 1976, he won on Bold Forbes and in 1985, on Spend A Buck. In 1976, Cordero won the Belmont Stakes on Bold Forbes, and he won the Preakness Stakes twice, once in 1980 aboard Codex (his winning status was not overturned despite widely-publicized accusations of illegal block-maneuvering and face-lashing of opponent-filly Genuine Risk with his whip) and the second time in 1984, aboard Gate Dancer.