Willingham notched a solo home run in the fourth inning and then launched a two-run bomb into the upper deck in the eighth inning that proved to be the game winner.
Oakland remained two games ahead of the second-place Texas Rangers, who lost to Pittsburgh Tuesday, in the AL West.
Oswaldo Arcia also homered for the Twins. Anthony Swarzak (2-2) picked up the win in relief for the Twins, who have won two in a row. Glen Perkins chalked up his 34th save with a perfect ninth inning.
Ryan Cook (6-4) was the victim of Willingham's eight-inning homer and was hung with the loss. Alberto Callaspo drove in two runs for the A's with his seventh home run, giving Oakland a 2-1 lead.
Notable deaths of 2014 [PHOTOS]