Anderson (2-0), making his second start of the season following his return from elbow ligament replacement surgery, dominated the Indians on two hits, while walking two and striking out five over seven innings.
The left-hander has yielded just one run over 14 frames since his return to the A's rotation.
Ryan Cook tossed a 1-2-3 eighth and Grant Balfour struck out three en route to his 15th save as Oakland -- which came into the game tied for the second American League wild card spot -- won for the ninth time in 11 games.
Roberto Hernandez (0-3) started for the Tribe and left after 2 1/3 innings with a right ankle sprain. He allowed three runs and one walk.