The Scunthorpe-born golfer was faced with a Hazeltine course deemed as "difficult" by Jack Nicklaus and at 7, 151 yards also carried the tag of being the second longest in Open history.
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 Hazeltine National 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 Hazeltine National Golf Club in Chaska, Minnesota.