The Twins put a runner on third with none away in the 10th and could not score, but they did not waste their chance in the 11th, while snapping a two-game skid.
Mauer doubled on a 3-2 pitch off Jonathan Broxton (1-2) to open the 11th and came in on Willingham's single to right.
That gave Casey Fien (1-0) the first victory of his major league career. Fien worked the 10th inning in his 16th relief appearance scattered over parts of three seasons.
His only other big league decision was a loss while pitching for Detroit in 2009.
Glen Perkins gave up a leadoff double in the bottom of the 11th, but worked out of the jam to pick up his fifth save.
Dennis Rodman pledges to end trips to North Korea
Susan Sarandon 'very excited' about daughter's pregnancy