Kotchman's single with one out in the sixth inning drove in Michael Brantley, breaking a 3-3 tie. Kotchman came around to score an insurance run when Jose Lopez followed with a double to chase Tigers starter Rick Porcello (3-4).
Travis Hafner added a pair of RBI for the American League Central-leading Indians, who notched their sixth win in eight games and took the opener of a three-game series against division rival Detroit.
Ubaldo Jimenez (5-3) worked around six walks to earn the victory, surrendering three runs on five hits in a six-inning stint.
Four Indians relievers closed the door the rest of the way, with Chris Perez racking up his 14th save despite allowing a walk and a hit in the ninth.
Alex Avila knocked a three-run homer for the Tigers, who fell three games behind the Indians in the AL Central.