The Indians, winners of eight in a row, and Rays both have 90-70 records. They own a one-game lead over Texas in the three-way chase for the two wild card positions in the American League.
It was announced Friday that if there is a three-way tie for those two spots at the end of the regular season, Cleveland will host Tampa Bay on Monday with the winner of that game qualifying for the playoffs.
The loser would then travel to Texas for a game on Tuesday and the winner of that one would join Monday's winner in the wild card contest on Wednesday.
Ryan Raburn highlighted a three-run second with a two-run double to give Cleveland a 7-0 cushion.
Minnesota closed its deficit to three runs through six innings, but the comeback ran out of steam under the brunt of a 17-hit Cleveland attack.
Corey Kluber (11-6) gave up six runs on 10 hits but still picked up the win by surviving 5 1/3 innings. Pedro Hernandez (3-3) was roughed up for six runs on seven hits in just 1 2/3 to take the loss.