Lackey (3-4) turned in his second straight outstanding performance, having given up a lone hit in a win against Minnesota last Sunday before leaving the game after a rain delay in the sixth. He limited the Indians to one unearned run while striking out eighth.
Carp's home run, his third, came in the second inning to give Lackey the lead.
Justin Masterson (7-3) allowed five runs on six hits in five innings to take the loss.
The Indians slipped out of first place in the American League Central, replaced by Detroit and its four-game winning streak.
Notable deaths of 2014 [PHOTOS]