Willingham opened the scoring with an RBI double in the first inning and made it 4-0 for Minnesota with an RBI single in the second.
After the Athletics had narrowed the gap to 4-2, Mauer came through with a two-run single in the fourth to ice the contest.
Darin Mastroianni knocked in a pair of runs for the Twins, who took the opener of a three-game series.
Brian Duensing (3-8) broke a personal three-start winless streak, limiting the A's to two runs (one earned) on five hits over six innings. The left-hander walked one and struck out seven.
Oakland's Brandon McCarthy (6-5) was tagged for six runs on 10 hits over 3 1/3 innings.
Josh Reddick and Josh Donaldson each drove in a run for the A's, who remained a half game out of an American League wild-card spot.