Cleveland captured the opener 4-0 and the sweep allowed the Indians to equal their longest winning streak of the season at six games.
Brantley tripled home Lou Marson to tie the score in the second game at 3-3 in the fifth inning, and he then scored on Crowe's perfect suicide squeeze.
Josh Tomlin (6-4) lasted five innings to get the win, surrendering three runs on nine hits.
The bullpen gave up a hit over the final four innings and Chris Perez recorded the last four outs for his 23rd save.
Justin Verlander (18-9), who had won his previous four decisions, took the loss. He gave up four runs on seven hits and struck out 10 in seven innings.
Detroit has dropped three in a row after a four-game winning streak.