Shin-Soo Choo opened the bottom of the 10th with a double off Twins' reliever Anthony Swarzak (3-5). He advanced to third on a grounder but Minnesota intentionally walked Carlos Santana and Michael Brantley to set up a force at home.
Kotchman bounced the second pitch he saw high over Swarzak and up the middle, sending Choo home with the game-winner.
Esmil Rogers (3-3), the fifth Cleveland pitcher, pitched a 1-2-3 10th and picked up the win.
The Twins, who had just six hits in the game, got their first run when Ben Revere scored on a wild pitch in the fourth. Cleveland answered with consecutive RBI singles from Brent Lillibridge and Jack Hannahan.
Minnesota went back up with runs in the fifth and sixth as Denard Span doubled in one and Trevor Plouffe made the score 3-2 in the sixth with his 23rd home run of the season.
The Indians tied the score with a two-out rally in the bottom of the inning as Kotchman doubled and came around to score on another Hannahan single.