Ramsay, 23, who played alongside Mickelson in the Masters in April, became the first British US Amateur champion in 95 years when he beat American John Kelly at HazeltineNational in August 2006.
The 37-year-old was two shots adrift of overnight leader Woods but kept his cool to card 70 to the American's 75 to win by three on the final green at the HazeltineNational Golf Club in Chaska, Minnesota.