After graduating from Mount Albert Grammar School (Auckland), Snell trained under Arthur Lydiard, who stressed running long distances in training to build endurance. He burst onto the international scene at the 1960 Olympic Games in Rome: the 800-metre race was supposed to be a showdown between Roger Moens of Belgium, the world-record holder, and George Kerr of Jamaica, but Snell shocked the field by charging past Moens in the last 25 metres to win by two-tenths of a second. In 1962 Snell set the 800-metre world record (1 min 44.3 sec), the world record for the 880-yard race (1 min 45.1 sec), and the world record for the mile (3 min 54.4 sec), which he lowered in 1964 (3 min 54.1 sec).

At the 1964 Olympic Games in Tokyo, Snell won the gold medals in both the 800- and 1,500-metre races—a rare feat. In the finals of the 800, finding himself boxed in against the rail with 250 metres to go, Snell dropped back to maneuver around the field before passing the leader, Kenyan Wilson Kiprugut, to win with a time (1 min 45.1 sec) bested only by his own world record. By the time he reached the finals of the 1,500-metre run, Snell was running his sixth race in eight days. With a lap to go, Snell was once again boxed in. This time, however, he simply raised his arm, and England's John Whetton gave him room to move. Snell broke free from the pack and cruised to his second gold medal of the 1964 Games.

In 1965 Snell retired from competitive racing; his autobiography, No Bugles, No Drums, was published that year. He subsequently began a career in sports physiology, earning degrees at the University of California at Davis (B.S.) and Washington State University (Ph.D.), and in 1981 he joined the staff at the University of Texas Southwestern Medical Center at Dallas. Snell was awarded the Member of the British Empire (MBE) in 1962.