The LA Dodgers have won 8 Cy Young Awards and the Dodgers franchise have won 9 total.
Brooklyn Dodgers 1956 - Don Newcombe
Los Angeles Dodgers 1962 - Don Drysdale
1963 - Sandy Koufax
1965 - Sandy Koufax
1966 - Sandy Koufax
1974 - Mike Marshall
1980 - Fernando Valenzuela
1988 - Orel Hershiser
2003 - Eric Gagne
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Don Newcombe of the Brooklyn Dodgers won Rookie of the Year in 1949 and both the MVP and Cy Young awards in 1956.
The Los Angeles Dodgers have won 10 Cy Young Awards (one as the Brooklyn Dodgers). Philadelphia and Atlanta are tied for second with 7. In the American league Boston and Baltimore are tied for first with 6. New York and Oakland each have 5. (source Baseball Reference.com)
Tom Glavine has won two Cy Young Awards, one in 1991, and the other in 1998.
Koufax won three Cy Young Awards, 1963, 1965 and 1966. Starting in 1956 when the Cy Young Award was created, only one award was presented for the entire Major Leagues. After 1966, both the American and National Leagues presented Cy Young Awards.
Roger Clemens won seven Cy Young Awards during his career.