Trailing 2-0 in the sixth, the Indians rallied for three runs to grab control.
Carlos Santana then recorded another double and Mark Reynolds was intentionally walked to get to Brantley. But the Tribe left fielder crossed up the strategy by grounding a 1-0 pitch up the middle for a decisive two-run single.
Brantley later smacked a two-run homer for Cleveland, which took the opener of a four-game series.
Ubaldo Jimenez (6-4) managed to hold Baltimore to two runs on eight hits over 5 1/3 innings, earning the victory.
Britton (1-2) was touched for three runs on six hits over five full frames for Baltimore, which fell 2 1/2 games behind American League East-leading Boston with the loss.