Oakland won for the fourth time in five games and the Twins had a three-game winning streak come to an end.
Gaudin (2-1) struck out four and did not walk a batter while throwing 85 pitches. Alan Embre worked a perfect eighth inning and Huston Street collected his sixth save despite putting two runners on with one away in the ninth.
Boof Bonser (1-4) was the hard-luck loser, giving up three hits and two runs over six innings.
Sweeney singled in a run in the third and hit a solo homer in the eighth. He also scored the Athletics' first run in the second inning on Emil Brown's sacrifice fly.