(year - Best Picture vs Best Director)

2013 - 12 Years a Slave vs Gravity
2012 - Argo vs Life of Pi
2005 - Crash vs Brokeback Mountain
2002 - Chicago vs The Pianist
2000 - Gladiator vs Traffic
1998 - Shakespeare in Love vs Saving Private Ryan
1989 - Driving Miss Daisy vs Born on the Fourth of July
1981 - Chariots of Fire vs Reds
1972 - The Godfather vs Cabaret
1967 - In the Heat of the Night vs The Graduate
1956 - Around the World in 80 Days vs Giant
1952 - The Greatest Show on Earth vs The Quiet Man
1951 - An American in Paris vs A Place in the Sun
1949 - All the Kings Men vs A Letter To Three Wives
1948 - Hamlet vs The Treasure of the Sierra Madre
1940 - Rebecca vs The Grapes of Wrath
1937 - The Life of Emile Zola vs The Awful Truth
1936 - The Great Ziegfeld vs Mr. Deeds Goes to Town
1935 - Mutiny on the Bounty vs The Informer
1931/1932 - Grand Hotel vs Bad Girl
1930/1931 - Cimarron vs Skippy
1928/1929 - The Broadway Melody vs The Divine Lady
1927/1928 - Wings vs Seventh Heaven (best drama) and Two Arabian Knights (best comedy)

(all other years the same film won both)