Sabathia, trying to become the second 17-game winner in the majors this year, lost his second straight start and for the third time in five outings. His current slump has come after he ran off a streak of seven consecutive wins.
Tampa Bay produced five home runs overall against Sabathia (16-7), who allowed 10 hits with seven strikeouts in eight innings.
Kotchman led off the third with his eighth homer of the season and Shoppach followed with his seventh. After Sabathia struck out the next two batters, Damon hit a 1-1 pitch out to right for his 10th.
Elliot Johnson tacked on a solo shot in the fifth and Evan Longoria hit his 18th of the year in the eighth.
David Price (10-10) picked up the victory, surrendering a run on six hits in eight innings.