Slowey (11-5) held the A's hitless in notching his ninth straight start without a loss. He walked three and struck out five before being pulled by Twins Manager Ron Gardenhire to start the eighth.
The Twins right-hander, who missed his previous start due to tendinitis in his pitching elbow, threw 106 pitches, 64 of them for strikes.
Jon Rauch gave up two runs in the eighth before Matt Capps tossed a scoreless ninth inning to secure the win and give Minnesota a three-game series sweep of the A's and a three-game lead over the Chicago White Sox in the American League Central.
Thome's seventh-inning shot put the Twins up 4-0.
Oakland starter Vin Mazzaro (6-5) surrendered two runs on seven hits in six innings.